cp's OEIS Frontend

A349997 Numbers k such that the number of primes in any interval [j^2,(j+1)^2], j>k, is not less than the number of primes in the interval [k^2,(k+1)^2].

%I A349997 #19 Aug 15 2025 12:57:40
%S A349997 1,7,11,17,18,26,27,32,46,50,56,58,85,88,92,137,143,145,152,157,178,
%T A349997 188,194,200,201,208,225,232,253,263,279,297,327,331,339,360,433,451,
%U A349997 485,506,536,541,607,696,708,717,768,799,801,806,904,913,1015,1059,1110,1111
%N A349997 Numbers k such that the number of primes in any interval [j^2,(j+1)^2], j>k, is not less than the number of primes in the interval [k^2,(k+1)^2].
%C A349997 All terms are empirical subject to the validity of Legendre's conjecture and the boundedness of the scatter band of A014085. See there for further information.
%H A349997 Hugo Pfoertner, <a href="/A349997/b349997.txt">Table of n, a(n) for n = 1..2414</a>
%H A349997 Joel E. Cohen, <a href="https://arxiv.org/abs/2508.08335">Conjectures about Primes and Cyclic Numbers</a>, arXiv:2508.08335 [math.NT], 2025. See pp. 9-10.
%F A349997 A014085(k) >= A014085(a(n)) for all k >= a(n).
%e A349997 a(1)=1: the interval [1^2, 2^2] contains A349999(1)=2 primes {2, 3}, and no later interval contains less than 2 primes.
%e A349997 a(2)=7: the interval [7^2, 8^2] contains A349999(2)=3 primes {53, 59, 61}, and no later interval contains less than 3 primes.
%e A349997 a(12)=58: the interval [58^2, 59^2] contains A349999(12)=13 primes {3371, ..., 3469}, and no later interval contains less than 13 primes.
%e A349997 a(13)=85: the interval [85^2, 86^2] contains A349999(13)=16 primes {7229, ..., 7393}, and no later interval contains less than 16 primes.
%Y A349997 Cf. A014085, A349998, A349999.
%K A349997 nonn
%O A349997 1,2
%A A349997 _Hugo Pfoertner_, Dec 09 2021