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].

1, 7, 11, 17, 18, 26, 27, 32, 46, 50, 56, 58, 85, 88, 92, 137, 143, 145, 152, 157, 178, 188, 194, 200, 201, 208, 225, 232, 253, 263, 279, 297, 327, 331, 339, 360, 433, 451, 485, 506, 536, 541, 607, 696, 708, 717, 768, 799, 801, 806, 904, 913, 1015, 1059, 1110, 1111

Offset: 1

Author

Hugo Pfoertner, Dec 09 2021

Keywords

• nonn

Comments

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.

Examples

			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.
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.
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.
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.
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..2414
• Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See pp. 9-10.

Crossrefs

Cf. A014085, A349998, A349999.

Formula

A014085(k) >= A014085(a(n)) for all k >= a(n).