A395513 - OEIS

%I #13 May 16 2026 18:18:33

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

%T 3852,448

%N 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.

%C 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.

%e 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.

%e 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].

%Y Cf. A025487, A046523.

%K nonn,more

%O 1,2

%A _David A. Corneth_, Apr 26 2026