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Schnirelmann density
In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Soviet mathematician Lev Schnirelmann, who was the first to study it.
The Schnirelmann density of a set of natural numbers A is defined as
where A(n) denotes the number of elements of A not exceeding n and inf is infimum.
The Schnirelmann density is well-defined even if the limit of A(n)/n as n → ∞ fails to exist (see upper and lower asymptotic density).
By definition, 0 ≤ A(n) ≤ n and n σA ≤ A(n) for all n, and therefore 0 ≤ σA ≤ 1, and σA = 1 if and only if A = N. Furthermore,
The Schnirelmann density is sensitive to the first values of a set:
In particular,
and
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Schnirelmann density AI simulator
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Schnirelmann density
In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Soviet mathematician Lev Schnirelmann, who was the first to study it.
The Schnirelmann density of a set of natural numbers A is defined as
where A(n) denotes the number of elements of A not exceeding n and inf is infimum.
The Schnirelmann density is well-defined even if the limit of A(n)/n as n → ∞ fails to exist (see upper and lower asymptotic density).
By definition, 0 ≤ A(n) ≤ n and n σA ≤ A(n) for all n, and therefore 0 ≤ σA ≤ 1, and σA = 1 if and only if A = N. Furthermore,
The Schnirelmann density is sensitive to the first values of a set:
In particular,
and