A395513 a(n) is the smallest number where the largest density of numbers with the same signature as A025487(n) among the first a(n) positive integers occurs.

1, 3, 4, 95, 8, 76, 16, 56, 7109105, 36, 208, 12645, 64, 108, 224, 27512, 128, 400, 3852, 448

Offset

1,2

Comments

Let Q(s, n) be the number of numbers <= n with prime signature s. Then a(n) is the (conjectured) smallest value k that maximizes the ratio Q(s, k)/k.

Links

Table of n, a(n) for n=1..20.

Example

a(1) = 1 as A025487(1) = 1 which has prime signature []. 1 is the only number in the positive integers with this signature. Therefore the ratio Q([], k)/k = 1/k is maximal when k = 1.
a(2) = 3 as A025487(2) = 2 which has prime signature [1]. Numbers with this signature are the primes. The largest density of primes in [1..k] is at k = 3. In the interval [1..3] there are two primes, 2 and 3. That is 2/3 of numbers in that interval. For any k > 3 the density of primes in the interval [1..k] is no more than 2/3 as at most every other number is prime hence has signature [1].

Crossrefs

Cf. A025487, A046523.

Sequence in context: A135237 A316197 A041253 * A297119 A156182 A126578

Adjacent sequences: A395510 A395511 A395512 * A395514 A395515 A395516

Keyword

nonn,more

Author

David A. Corneth, Apr 26 2026

Status

approved