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700 (number)
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| ||||
|---|---|---|---|---|
| Cardinal | seven hundred | |||
| Ordinal | 700th (seven hundredth) | |||
| Factorization | 22 × 52 × 7 | |||
| Greek numeral | Ψ´ | |||
| Roman numeral | DCC, dcc | |||
| Binary | 10101111002 | |||
| Ternary | 2212213 | |||
| Senary | 31246 | |||
| Octal | 12748 | |||
| Duodecimal | 4A412 | |||
| Hexadecimal | 2BC16 | |||
| Armenian | Չ | |||
| Hebrew | ת"ש / ן | |||
| Babylonian cuneiform | 𒌋𒐕𒐏 | |||
| Egyptian hieroglyph | 𓍨 | |||
700 (seven hundred) is the natural number following 699 and preceding 701.
It is the sum of four consecutive primes (167 + 173 + 179 + 181), the perimeter of a Pythagorean triangle (75 + 308 + 317)[1] and a Harshad number.
Integers from 701 to 799
[edit]Nearly all of the palindromic integers between 700 and 800 (i.e. nearly all numbers in this range that have both the hundreds and units digit be 7) are used as model numbers for Boeing Commercial Airplanes.
700s
[edit]- 701 = prime number, sum of three consecutive primes (229 + 233 + 239), Chen prime, Eisenstein prime with no imaginary part
- 702 = 2 × 33 × 13, pronic number,[2] nontotient, Harshad number
- 703 = 19 × 37, the 37th triangular number,[3] a hexagonal number,[4] smallest number requiring 73 fifth powers for Waring representation, Kaprekar number,[5] area code for Northern Virginia along with 571, a number commonly found in the formula for body mass index
- 704 = 26 × 11, Harshad number, lazy caterer number (sequence A000124 in the OEIS), area code for the Charlotte, NC area.
- 705 = 3 × 5 × 47, sphenic number, smallest Bruckman-Lucas pseudoprime (sequence A005845 in the OEIS)
- 706 = 2 × 353, nontotient, Smith number[6]
- 707 = 7 × 101, sum of five consecutive primes (131 + 137 + 139 + 149 + 151), palindromic number, number of lattice paths from (0,0) to (5,5) with steps (0,1), (1,0) and, when on the diagonal, (1,1).[7]
- 708 = 22 × 3 × 59, number of partitions of 28 that do not contain 1 as a part[8]
- 709 = prime number; happy number. It is the seventh in the series 2, 3, 5, 11, 31, 127, 709 where each number is the nth prime with n being the number preceding it in the series, therefore, it is a prime index number.
710s
[edit]- 710 = 2 × 5 × 71, sphenic number, nontotient, number of forests with 11 vertices[9][10]
- 711 = 32 × 79, Harshad number, number of planar Berge perfect graphs on 7 nodes.[11] Also the phone number of Telecommunications Relay Service, commonly used by the deaf and hard-of-hearing.
- 712 = 23 × 89, refactorable number, sum of the first twenty-one primes, totient sum for first 48 integers. It is the largest known number such that it and its 8th power (66,045,000,696,445,844,586,496) have no common digits.
- 713 = 23 × 31, Blum integer, main area code for Houston, TX. In Judaism there are 713 letters on a Mezuzah scroll.
- 714 = 2 × 3 × 7 × 17, sum of twelve consecutive primes (37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83), nontotient, balanced number,[12] member of Ruth–Aaron pair (either definition); area code for Orange County, California.
- Flight 714 to Sidney is a Tintin graphic novel.
- 714 is the badge number of Sergeant Joe Friday.
- 715 = 5 × 11 × 13, sphenic number, pentagonal number,[13] pentatope number ( binomial coefficient ),[14] Harshad number, member of Ruth-Aaron pair (either definition)
- The product of 714 and 715 is the product of the first 7 prime numbers (2, 3, 5, 7, 11, 13, and 17)
- 716 = 22 × 179, area code for Buffalo, NY
- 717 = 3 × 239, palindromic number
- 718 = 2 × 359, area code for Brooklyn, NY and Bronx, NY
- 719 = prime number, factorial prime (6! − 1),[15] Sophie Germain prime,[16] safe prime,[17] sum of seven consecutive primes (89 + 97 + 101 + 103 + 107 + 109 + 113), Chen prime, Eisenstein prime with no imaginary part
720s
[edit]- 720 = 24 × 32 × 5.
- 6 factorial, highly composite number, Harshad number in every base from binary to decimal, highly totient number.
- two round angles (= 2 × 360).
- five gross (= 500 duodecimal, 5 × 144).
- 241-gonal number.
- 721 = 7 × 103, sum of nine consecutive primes (61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101), centered hexagonal number,[18] smallest number that is the difference of two positive cubes in two ways,
- 722 = 2 × 192, nontotient, number of odd parts in all partitions of 15,[19] area of a square with diagonal 38[20]
- G.722 is a freely available file format for audio file compression. The files are often named with the extension "722".
- 723 = 3 × 241, side length of an almost-equilateral Heronian triangle[21]
- 724 = 22 × 181, sum of four consecutive primes (173 + 179 + 181 + 191), sum of six consecutive primes (107 + 109 + 113 + 127 + 131 + 137), nontotient, side length of an almost-equilateral Heronian triangle,[22] the number of n-queens problem solutions for n = 10,
- 725 = 52 × 29, side length of an almost-equilateral Heronian triangle[23]
- 726 = 2 × 3 × 112, pentagonal pyramidal number[24]
- 727 = prime number, palindromic prime, lucky prime,[25]
- 728 = 23 × 7 × 13, nontotient, Smith number,[6] cabtaxi number,[26] 728!! - 1 is prime,[27] number of cubes of edge length 1 required to make a hollow cube of edge length 12, 72864 + 1 is prime, number of connected graphs on 5 labelled vertices
- 729 = 272 = 93 = 36.
- the square of 27, and the cube of 9, the sixth power of three, and because of these properties, a perfect totient number.[28]
- centered octagonal number,[29] Smith number[6]
- the number of times a philosopher king's pleasure is greater than a tyrant's pleasure according to Plato in the Republic
- the largest three-digit cube. (9 x 9 x 9)
- the only three-digit sixth power. (3 x 3 x 3 x 3 x 3 x 3)
730s
[edit]- 730 = 2 × 5 × 73, sphenic number, nontotient, Harshad number, number of generalized weak orders on 5 points [30]
- 731 = 17 × 43, sum of three consecutive primes (239 + 241 + 251), number of Euler trees with total weight 7 [31]
- 732 = 22 × 3 × 61, sum of eight consecutive primes (73 + 79 + 83 + 89 + 97 + 101 + 103 + 107), sum of ten consecutive primes (53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97), Harshad number, number of collections of subsets of {1, 2, 3, 4} that are closed under union and intersection [32]
- 733 = prime number, emirp, balanced prime,[33] permutable prime, sum of five consecutive primes (137 + 139 + 149 + 151 + 157)
- 734 = 2 × 367, nontotient, number of traceable graphs on 7 nodes [34]
- 735 = 3 × 5 × 72, Harshad number, Zuckerman number, smallest number such that uses same digits as its distinct prime factors
- 736 = 25 × 23, centered heptagonal number,[35] happy number, nice Friedman number since 736 = 7 + 36, Harshad number
- 737 = 11 × 67, palindromic number, blum integer.
- 738 = 2 × 32 × 41, Harshad number.
- 739 = prime number, strictly non-palindromic number,[36] lucky prime,[25] happy number, prime index prime
740s
[edit]- 740 = 22 × 5 × 37, nontotient, number of connected squarefree graphs on 9 nodes [37]
- 741 = 3 × 13 × 19, sphenic number, 38th triangular number[3]
- 742 = 2 × 7 × 53, sphenic number, decagonal number,[38] icosahedral number. It is the smallest number that is one more than triple its reverse. Lazy caterer number (sequence A000124 in the OEIS). Number of partitions of 30 into divisors of 30.[39]
- 743 = prime number, Sophie Germain prime, Chen prime, Eisenstein prime with no imaginary part
- 744 = 23 × 3 × 31, sum of four consecutive primes (179 + 181 + 191 + 193). It is the coefficient of the first degree term of the expansion of Klein's j-invariant, and the zeroth degree term of the Laurent series of the J-invariant. Furthermore, 744 = 3 × 248 where 248 is the dimension of the Lie algebra E8.
- 745 = 5 × 149 = 24 + 36, number of non-connected simple labeled graphs covering 6 vertices[40]
- 746 = 2 × 373 = 15 + 24 + 36 = 17 + 24 + 36, nontotient, number of non-normal semi-magic squares with sum of entries equal to 6[41]
- 747 = 32 × 83 = ,[42] palindromic number.
- 748 = 22 × 11 × 17, nontotient, happy number, primitive abundant number[43]
- 749 = 7 × 107, sum of three consecutive primes (241 + 251 + 257), blum integer
750s
[edit]- 750 = 2 × 3 × 53, enneagonal number.[44]
- 751 = prime number with a prime number of prime digits,[45] Chen prime, emirp,
- 752 = 24 × 47, nontotient, number of partitions of 11 into parts of 2 kinds[46]
- 753 = 3 × 251, blum integer
- 754 = 2 × 13 × 29, sphenic number, nontotient, totient sum for first 49 integers, number of different ways to divide a 10 × 10 square into sub-squares [47]
- 755 = 5 × 151, number of vertices in a regular drawing of the complete bipartite graph K9,9.[48]
- 756 = 22 × 33 × 7, sum of six consecutive primes (109 + 113 + 127 + 131 + 137 + 139), pronic number,[2] Harshad number
- 757 = prime number, palindromic prime, sum of seven consecutive primes (97 + 101 + 103 + 107 + 109 + 113 + 127), happy number.
- "The 757" is a local nickname for the Hampton Roads area in the U.S. state of Virginia, derived from the telephone area code that covers almost all of the metropolitan area
- 758 = 2 × 379, nontotient, prime number of measurement [49]
- 759 = 3 × 11 × 23, sphenic number, sum of five consecutive primes (139 + 149 + 151 + 157 + 163), a q-Fibonacci number for q=3 [50]
760s
[edit]- 760 = 23 × 5 × 19, centered triangular number,[51] number of fixed heptominoes.
- 761 = prime number, emirp, Sophie Germain prime,[16] Chen prime, Eisenstein prime with no imaginary part, centered square number[52]
- 762 = 2 × 3 × 127, sphenic number, sum of four consecutive primes (181 + 191 + 193 + 197), nontotient, Smith number,[6] admirable number, number of 1's in all partitions of 25 into odd parts,[53] see also Six nines in pi
- 763 = 7 × 109, sum of nine consecutive primes (67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103), number of degree-8 permutations of order exactly 2 [54]
- 764 = 22 × 191, telephone number[55]
- 765 = 32 × 5 × 17, octagonal pyramidal number [56]
- 766 = 2 × 383, centered pentagonal number,[57] nontotient, sum of twelve consecutive primes (41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89)
- 767 = 13 × 59, Thabit number (28 × 3 − 1), palindromic number.
- 768 = 28 × 3,[58] sum of eight consecutive primes (79 + 83 + 89 + 97 + 101 + 103 + 107 + 109)
- 769 = prime number, Chen prime, lucky prime,[25] Proth prime[59]
770s
[edit]- 770 = 2 × 5 × 7 × 11, nontotient, Harshad number
- is prime[60]
- Famous room party in New Orleans hotel room 770, giving the name to a well known science fiction fanzine called File 770
- Holds special importance in the Chabad-Lubavitch Hasidic movement.
- 771 = 3 × 257, sum of three consecutive primes in arithmetic progression (251 + 257 + 263). Since 771 is the product of the distinct Fermat primes 3 and 257, a regular polygon with 771 sides can be constructed using compass and straightedge, and can be written in terms of square roots.
- 772 = 22 × 193, 772!!!!!!+1 is prime[61]
- 773 = prime number, Eisenstein prime with no imaginary part, tetranacci number,[62] prime index prime, sum of the number of cells that make up the convex, regular 4-polytopes
- 774 = 2 × 32 × 43, nontotient, totient sum for first 50 integers, Harshad number
- 775 = 52 × 31, member of the Mian–Chowla sequence[63]
- 776 = 23 × 97, refactorable number, number of compositions of 6 whose parts equal to q can be of q2 kinds[64]
- 777 = 3 × 7 × 37, sphenic number, Harshad number, palindromic number, 3333 in senary (base 6) counting.
- 778 = 2 × 389, nontotient, Smith number[6]
- 779 = 19 × 41, highly cototient number[67]
780s
[edit]- 780 = 22 × 3 × 5 × 13, sum of four consecutive primes in a quadruplet (191, 193, 197, and 199); sum of ten consecutive primes (59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101), 39th triangular number,[3]a hexagonal number,[4] Harshad number
- 780 and 990 are the fourth smallest pair of triangular numbers whose sum and difference (1770 and 210) are also triangular.
- 781 = 11 × 71. 781 is the sum of powers of 5/repdigit in base 5 (11111), Mertens function(781) = 0, lazy caterer number (sequence A000124 in the OEIS)
- 782 = 2 × 17 × 23, sphenic number, nontotient, pentagonal number,[13] Harshad number, also, 782 gear used by U.S. Marines
- 783 = 33 × 29, heptagonal number
- 784 = 24 × 72 = 282 = , the sum of the cubes of the first seven positive integers, happy number
- 785 = 5 × 157, Mertens function(785) = 0, number of series-reduced planted trees with 6 leaves of 2 colors [68]
- 786 = 2 × 3 × 131, sphenic number, admirable number. See also its use in Muslim numerological symbolism.
- 787 = prime number, sum of five consecutive primes (149 + 151 + 157 + 163 + 167), Chen prime, lucky prime,[25] palindromic prime.
- 788 = 22 × 197, nontotient, number of compositions of 12 into parts with distinct multiplicities [69]
- 789 = 3 × 263, sum of three consecutive primes (257 + 263 + 269), Blum integer
790s
[edit]- 790 = 2 × 5 × 79, sphenic number, nontotient, a Harshad number in bases 2, 7, 14 and 16, an aspiring number,[70] the aliquot sum of 1574.
- 791 = 7 × 113, centered tetrahedral number, sum of the first twenty-two primes, sum of seven consecutive primes (101 + 103 + 107 + 109 + 113 + 127 + 131)
- 792 = 23 × 32 × 11, number of integer partitions of 21,[71] binomial coefficient , Harshad number, sum of the nontriangular numbers between successive triangular numbers
- 793 = 13 × 61, Mertens function(793) = 0, star number,[72] happy number
- 794 = 2 × 397 = 16 + 26 + 36,[73] nontotient
- 795 = 3 × 5 × 53, sphenic number, Mertens function(795) = 0, number of permutations of length 7 with 2 consecutive ascending pairs [74]
- 796 = 22 × 199, sum of six consecutive primes (113 + 127 + 131 + 137 + 139 + 149), Mertens function(796) = 0
- 797 = prime number, Chen prime, Eisenstein prime with no imaginary part, palindromic prime, two-sided prime, prime index prime.
- 798 = 2 × 3 × 7 × 19, Mertens function(798) = 0, nontotient, product of primes indexed by the prime exponents of 10! [75]
- 799 = 17 × 47, smallest number with digit sum 25 [76]
References
[edit]- ^ Sloane, N. J. A. (ed.). "Sequence A024364 (Ordered perimeters of primitive Pythagorean triangles)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2022.
- ^ a b "Sloane's A002378 : Oblong (or promic, pronic, or heteromecic) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ a b c "Sloane's A000217 : Triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ a b "Sloane's A000384 : Hexagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ "Sloane's A006886 : Kaprekar numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ a b c d e "Sloane's A006753 : Smith numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ Sloane, N. J. A. (ed.). "Sequence A026671 (Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (1,1))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A002865 (Number of partitions of n that do not contain 1 as a part)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2 June 2022.
- ^ Hougardy, Stefan (October 2006). "Classes of perfect graphs". Discrete Mathematics. 306 (19–20): 2529–2571. doi:10.1016/j.disc.2006.05.021.
- ^ Sloane, N. J. A. (ed.). "Sequence A005195 (Number of forests with n unlabeled nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A123449 (Number of planar Berge perfect graphs on n nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A020492 (Balanced numbers: numbers k such that phi(k) (A000010) divides sigma(k) (A000203))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b "Sloane's A000326 : Pentagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ "Sloane's A000332 : Binomial coefficient binomial(n,4)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ "Sloane's A088054 : Factorial primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ a b "Sloane's A005384 : Sophie Germain primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ "Sloane's A005385 : Safe primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ Sloane, N. J. A. (ed.). "Sequence A066897 (Total number of odd parts in all partitions of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A001105 (a(n) = 2*n^2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A016064 (Smallest side lengths of almost-equilateral Heronian triangles)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A003500 (a(n) = 4*a(n-1) - a(n-2) with a(0) = 2, a(1) = 4)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A335025 (Largest side lengths of almost-equilateral Heronian triangles)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 May 2022.
- ^ "Sloane's A002411 : Pentagonal pyramidal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ a b c d "Sloane's A031157 : Numbers that are both lucky and prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ "Sloane's A047696 : Smallest positive number that can be written in n ways as a sum of two (not necessarily positive) cubes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ Sloane, N. J. A. (ed.). "Sequence A007749 (Numbers k such that k!! - 1 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 May 2022.
- ^ "Sloane's A082897 : Perfect totient numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ "Sloane's A016754 : Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ Sloane, N. J. A. (ed.). "Sequence A004123 (Number of generalized weak orders on n points)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A007317 (Binomial transform of Catalan numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A306445 (Number of collections of subsets of {1, 2, ..., n} that are closed under union and intersection)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 May 2022.
- ^ "Sloane's A006562 : Balanced primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ Sloane, N. J. A. (ed.). "Sequence A057864 (Number of simple traceable graphs on n nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 May 2022.
- ^ "Sloane's A069099 : Centered heptagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ "Sloane's A016038 : Strictly non-palindromic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ Sloane, N. J. A. (ed.). "Sequence A077269 (Number of connected squarefree graphs on n nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 23 May 2022.
- ^ "Sloane's A001107 : 10-gonal (or decagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ Sloane, N. J. A. (ed.). "Sequence A018818 (Number of partitions of n into divisors of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A327070 (Number of non-connected simple labeled graphs covering n vertices)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 23 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A321719 (Number of non-normal semi-magic squares with sum of entries equal to n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 30 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A064628 (Floor(4^n / 3^n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 30 May 2022.
- ^ "Sloane's A091191 : Primitive abundant numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ "Sloane's A001106 : 9-gonal (or enneagonal or nonagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ Sloane, N. J. A. (ed.). "Sequence A348699 (Primes with a prime number of prime digits)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A000712 (Generating function = Product_{m>=1} 1/(1 - x^m)^2; a(n) = number of partitions of n into parts of 2 kinds)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 30 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A034295 (Number of different ways to divide an n X n square into sub-squares)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 23 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A331755 (Number of vertices in a regular drawing of the complete bipartite graph K_{9,9})". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 23 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A002049 (Prime numbers of measurement)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 23 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A015474 (q-Fibonacci numbers for q=3)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 23 May 2022.
- ^ "Sloane's A005448 : Centered triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ "Sloane's A001844 : Centered square numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ Sloane, N. J. A. (ed.). "Sequence A036469 (Partial sums of A000009 (partitions into distinct parts))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A001189 (Number of degree-n permutations of order exactly 2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 23 May 2022.
- ^ "Sloane's A000085 : Number of self-inverse permutations on n letters, also known as involutions". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ Sloane, N. J. A. (ed.). "Sequence A002414 (Octagonal pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 23 May 2022.
- ^ "Sloane's A005891 : Centered pentagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ Sloane, N. J. A. (ed.). "Sequence A007283 (a(n) = 3*2^n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 30 May 2022.
- ^ "Sloane's A080076 : Proth primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ Sloane, N. J. A. (ed.). "Sequence A162862 (Numbers n such that n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 30 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A085150 (Numbers n such that n!!!!!!+1 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 30 May 2022.
- ^ "Sloane's A000078 : Tetranacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ "Sloane's A005282 : Mian-Chowla sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ (sequence A033453 in the OEIS)
- ^ Posner, Eliezer. "On the Meaning of Three". Chabad. Retrieved 2 July 2016.
- ^ Dennis, Geoffrey. "Judaism & Numbers". My Jewish Learning. Retrieved 2 July 2016.
- ^ "Sloane's A100827 : Highly cototient numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2016.
- ^ Sloane, N. J. A. (ed.). "Sequence A050381 (Number of series-reduced planted trees with n leaves of 2 colors)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A242882 (Number of compositions of n into parts with distinct multiplicities)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A063769 (Aspiring numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A000041 (a(n) = number of partitions of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A003154 (Centered 12-gonal numbers. Also star numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A001550 (a(n) = 1^n + 2^n + 3^n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A000274 (Number of permutations of length n with 2 consecutive ascending pairs)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A325508 (Product of primes indexed by the prime exponents of n!)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 May 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A051885 (Smallest number whose sum of digits is n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 May 2022.
700 (number)
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Mathematical Properties
700 belongs to several number-theoretic categories due to its factorization and other attributes. As a composite number, it is the product of three distinct primes each raised to a power.[1] Its divisors are 1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, and 700.[5] Notably, 700 is the number of symmetric octacubes (polycubes composed of 8 cubes with rotational symmetry) in combinatorial geometry.[6] It is not a perfect square (√700 ≈ 26.457), perfect cube, or Fibonacci number.Numeral Representations
Beyond Roman numerals, 700 in binary is 1010111100,[7] in hexadecimal 2BC, and in base-10 it is simply 700. In ancient numeral systems, such as Greek, it is denoted as ψʹ in the alphabetic system.[8]Historical and Cultural Contexts
The number 700 has appeared in historical timelines, such as the year 700 AD marking the flourishing of the Srivijaya kingdom in Sumatra.[9] In modern culture, it denotes metrics like the 700 MHz frequency band in telecommunications standards[10] or 700-calorie diets in nutrition, though these are contextual uses rather than inherent properties.Mathematical Properties
Prime Factorization
The prime factorization of 700 is obtained by repeatedly dividing by the smallest prime numbers until only primes remain. Starting with 700, divide by 2 to get 350; divide 350 by 2 to get 175; divide 175 by 5 to get 35; divide 35 by 5 to get 7; and 7 is prime. Thus, .[5][1] From this factorization, Euler's totient function counts the integers up to 700 that are coprime to 700. The formula is , where are the distinct prime factors. Substituting the values: . First, ; then ; finally, . Thus, .[11][12] The number of distinct prime factors, denoted , is 3 (corresponding to 2, 5, and 7). The total number of prime factors counting multiplicity, denoted , is 5 (two 2's, two 5's, and one 7).[13][11]Divisors and Multiples
The positive divisors of 700 are 1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, and 700.[14] These divisors arise from the prime factorization of 700 as , where every combination of the exponents (0 to 2 for 2 and 5, 0 to 1 for 7) yields a divisor.[15] The number of divisors, denoted or , is calculated using the formula for a number with prime factorization as . Thus, .[15][14] The sum of the divisors, denoted , is the total .[14] The sum of the proper divisors (excluding 700) is 1036. The multiples of 700 follow the pattern for positive integers . The first ten multiples are presented below for reference:| Multiple | |
|---|---|
| 1 | 700 |
| 2 | 1400 |
| 3 | 2100 |
| 4 | 2800 |
| 5 | 3500 |
| 6 | 4200 |
| 7 | 4900 |
| 8 | 5600 |
| 9 | 6300 |
| 10 | 7000 |
Other Characteristics
700 is an even number, as it is divisible by 2.[11] It is also a composite number, possessing divisors other than 1 and itself.[16] 700 qualifies as an abundant number, where the sum of its proper divisors totals 1036, exceeding the number itself; this contrasts with perfect numbers, where the sum equals the number, and deficient numbers, where it falls short.[17] The full sum of divisors, , is 1736, which is greater than twice 700 (1400), confirming its abundant status.[14] Additionally, 700 is a Harshad number, being divisible by the sum of its digits: 7 + 0 + 0 = 7, and 700 ÷ 7 = 100.[18] Under the Collatz conjecture, the sequence starting from 700 reaches 1 after 82 steps.[19] As the 700th positive integer, 700 falls between the squares of 26 (676) and 27 (729).[20]Representations in Numeral Systems
Decimal and Roman Numerals
In the decimal numeral system, 700 is written as 700, a three-digit number where the leftmost digit represents 7 hundreds (700), the middle digit 0 tens (0), and the rightmost digit 0 units (0), following the base-10 positional notation that assigns powers of 10 to each place value from right to left. This Hindu-Arabic numeral system, originating in India and transmitted through the Islamic world, was introduced to Europe by the Italian mathematician Leonardo Fibonacci in his 1202 book Liber Abaci, which popularized its use for arithmetic and commerce over the more cumbersome Roman system.[21] In English, the number is pronounced "seven hundred," combining the cardinal numeral "seven"—derived from Old English seofon, from Proto-Germanic sebun, ultimately tracing to Proto-Indo-European septḿ̥ meaning "seven"—with "hundred," from Old English hundred, a compound of hund ("a hundred," akin to the root for "hund-redth") and rad ("count" or "reckoning"), denoting a base unit of 100 in early Germanic reckoning.[22][23] The Roman numeral representation of 700 is DCC, constructed additively as D (500) plus CC (two instances of C, each worth 100, totaling 200), adhering to the standard rules of Roman notation where values are summed from left to right without subtractive principles for this quantity, as subtractive notation (e.g., CD for 400) applies only to specific cases like 4, 9, 40, 90, 400, and 900 to avoid excessive repetition.[24] Across languages, the verbal form of 700 varies: in French, it is "sept cents," literally "seven hundreds," reflecting the language's tendency to pluralize "cent" (hundred) for multiples beyond one.[25] In Mandarin Chinese, it is "qī bǎi" (七百), combining "qī" for seven and "bǎi" for hundred, written with the characters 七 (seven) and 百 (hundred) in the Sino-Xenic numbering system.[26]Binary, Octal, and Hexadecimal
In binary, the number 700 is represented as , requiring 10 bits to express its value.[27] This positional notation uses powers of 2, where the leftmost bit represents and the rightmost is . To convert 700 to binary, repeatedly divide by 2 and record the remainders from bottom to top: remainder 0, r 0, r 1, r 1, r 1, r 1, r 0, r 1, r 0, r 1, yielding .[28] The binary system forms the foundation of digital computing, as it aligns directly with the two-state logic of electronic circuits (0 for off, 1 for on).[29] In octal (base-8), 700 is represented as .[30] This can be derived from the binary form by grouping bits into sets of three from the right (padding with a leading zero if needed): becomes , which converts to since , , , and . Alternatively, divide repeatedly by 8: remainder 4, r 7, r 2, r 1, reading remainders upward as .[31] Octal representations compactly group three binary digits each, and were commonly used in early computing for file permissions and assembly language addressing.[32] In hexadecimal (base-16), 700 is represented as .[33] Digits beyond 9 use letters A-F (where A=10, B=11, C=12, etc.). Conversion involves repeated division by 16: remainder 12 (C), r 11 (B), r 2, yielding from bottom to top.[34] Hexadecimal efficiently encodes four binary digits per symbol, making it prevalent in modern computing for displaying memory addresses, debugging low-level code, and specifying colors in web development (e.g., RGB values).[29] The following table summarizes these representations for quick comparison:| Numeral System | Representation | Equivalent Decimal Value |
|---|---|---|
| Binary | 1010111100 | 700 |
| Octal | 1274 | 700 |
| Hexadecimal | 2BC | 700 |
Significance in Science and Technology
In Physics and Chemistry
In physics, the number 700 is notably associated with the electromagnetic spectrum, where 700 nanometers (nm) marks the approximate upper limit of visible light, corresponding to the deepest red hues perceivable by the human eye. The visible spectrum spans roughly from 400 nm (violet) to 700 nm (red), encompassing the wavelengths that stimulate human color vision.[35] This range is fundamental in optics and photometry, as it defines the portion of the electromagnetic spectrum interacting directly with biological photoreceptors.[36] The energy of a photon at this wavelength can be calculated using the formula , where is Planck's constant ( eV·s), is the speed of light ( m/s), and nm ( m). Substituting these values yields eV, a low-energy photon typical of red light used in applications like photosynthesis studies, where chlorophyll absorbs near 700 nm.[37][38] This energy level underscores the transition from visible to near-infrared radiation, influencing fields such as spectroscopy and photobiology.[39] In chemistry and materials science, 700°C serves as a critical temperature threshold in processes like sintering of ceramics, where powders are heated to achieve densification without full melting. For instance, certain dielectric ceramics form desired phases when processed at 700°C in methods like sol-gel synthesis, aiding in the development of materials with controlled microstructures. Similarly, in metallurgy, 700°C is employed in process annealing of low-alloy steels to relieve internal stresses and refine grain structure while avoiding excessive phase transformations.[40] These applications highlight 700's role in controlling material properties through controlled thermal treatments. Particle physics experiments frequently reference energies around 700 MeV (mega-electronvolts), such as in electron scattering studies on nuclei like carbon-12, where incident electron energies of 700 MeV probe proton spectral functions and nuclear structure via deep inelastic scattering.[41] Accelerators like the JINR Phasotron, which operates at 680 MeV, have historically enabled high-intensity proton beams for fragmentation cross-section measurements up to around 680 MeV/A, aiding understanding of cosmic ray interactions and radiation therapy.[42] These examples illustrate 700 MeV as a benchmark energy scale in probing subatomic interactions.In Computing and Engineering
In computing, the number 700 appears in non-standard extensions to HTTP status codes, where it is used by certain proxy servers like Squid to indicate soft errors, such as unexpected responses during authentication or proxy handling, beyond the standard range of codes up to 599 defined in RFC 9110.[43] These extensions allow proxies to signal internal issues without disrupting standard client-server interactions, though they are not part of the official IETF specifications. The IBM 700 series, introduced in the 1950s, represented a pivotal line of vacuum-tube-based mainframe computers that advanced commercial data processing and scientific computing. The series debuted with the IBM 701 in 1953, capable of up to 16,000 additions per second and featuring up to 4,096 words of electrostatic (Williams tube) storage memory, marking IBM's entry into high-speed electronic computing for engineering applications.[44] Subsequent models like the 704 and 709 built on this foundation, influencing transistor-based systems in the 1960s. In mobile communications engineering, the 700 MHz frequency band is allocated for LTE networks, particularly Band 12, which operates in the lower 700 MHz spectrum (uplink: 699–716 MHz, downlink: 729–746 MHz) to provide extended coverage in rural and indoor environments due to its favorable propagation characteristics.[45] This band supports up to 18 MHz of channel bandwidth per direction, enhancing signal penetration and range compared to higher frequencies. The bandwidth for such a paired frequency-division duplex (FDD) band is approximated as , where is the center frequency and is the lower edge, yielding about 34 MHz total for Band 12 to accommodate multiple carriers. Compact discs (CDs) commonly feature a capacity tied to 700 in storage contexts, where standard 74-minute audio CDs hold approximately 650–700 MB of data, depending on error correction and file system overhead. This equates to about 650 MB (650 × 10^6 bytes) for data modes under ISO/IEC 10149, but manufacturers often label 80-minute CD-R variants as 700 MB using decimal notation (700 × 10^6 bytes), which operating systems may display as roughly 668 MiB in binary (mebibytes) due to the distinction between base-10 and base-2 byte measurements. In hexadecimal representation, 700 decimal is 0x2BC, occasionally referenced in low-level memory addressing for legacy systems.Historical and Cultural Contexts
In History and Chronology
The year 700 marks a pivotal point in chronology as a round number signifying the conclusion of the 7th century AD, a era defined by profound geopolitical shifts, including the consolidation of Islamic rule in the Middle East and North Africa, the zenith of Chinese imperial expansion under the Tang Dynasty, and the reconfiguration of power in Europe and Byzantium, ushering in deeper medieval transitions.[46] This period's end underscored the transition from late antiquity to the early Middle Ages, with the rise of new empires disrupting longstanding Roman and Persian influences.[47] In the Anno Domini dating system, 700 AD fell during the reign of Umayyad Caliph Abd al-Malik (r. 685–705), whose administration stabilized the caliphate after civil strife through key reforms, such as standardizing the Arab-Sassanian silver dirham coinage in 696–697 and mandating Arabic for official documents by 700, enhancing imperial unity across diverse territories from Spain to Central Asia.[48] Concurrently, in East Asia, the Tang Dynasty under Empress Wu Zetian (r. 690–705) reached a high point of prosperity, with advancements in poetry, sculpture, and trade along the Silk Road, as exemplified by the completion of grand Buddhist monuments like the Longmen Grottoes.[49] These developments highlighted 700 AD as a nexus of cultural and administrative innovation in both Islamic and Chinese spheres. The date 700 AD was calculated under the Julian calendar, introduced by Julius Caesar in 45 BC and widely adopted in the Christian world by the 4th century AD, featuring a leap year that began on a Wednesday.[50] The subsequent Gregorian reform in 1582 AD, promulgated by Pope Gregory XIII to correct the Julian calendar's overestimation of the solar year by about 11 minutes annually, did not retroactively alter 700 AD dates, as the reform addressed accumulated drift over centuries without impacting earlier chronology.[51] Archaeological findings associate circa 700 BC with the onset of decline for the Urartu kingdom in the Armenian Highlands, where Assyrian campaigns under Sargon II (r. 721–705 BC) in the late 8th and early 7th centuries BC eroded its fortifications and economy, culminating in its collapse by the mid-6th century BC amid invasions by Scythians and Medes.[52] Likewise, around 700 BC, the Olmec civilization in Mesoamerica—centered at sites like San Lorenzo and La Venta—was transitioning from its formative phase (c. 1400–900 BC) to its apogee (c. 900–600 BC), marked by monumental architecture and jade carvings, before environmental factors and internal shifts initiated a gradual depopulation and cultural transformation by 400 BC.[53]| Period | Key Events |
|---|---|
| 700 AD | Umayyad Caliphate under Abd al-Malik implements currency reforms, unifying the economy across conquered lands from Iberia to India.[48] Tang Dynasty in China, ruled by Empress Wu Zetian, fosters artistic and religious patronage, including expansions in Buddhist art and literature.[49] |
| 700 BC | Neo-Assyrian Empire, under kings such as Esarhaddon (r. 681–669 BC) and Ashurbanipal (r. 669–627 BC), dominates the Near East, subjugating Phoenicia and Israel while pressuring Urartu.[54] Greek city-states expand colonization efforts, founding settlements like Cumae in Italy (c. 750 BC) and Cyrene in Libya (c. 631 BC), spreading Hellenic culture.[54] Olmec society in Mesoamerica produces iconic colossal heads and develops early writing systems during its peak phase at [La Venta](/page/La Venta).[53] |
