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A349998 Numbers k such that the number of primes in any interval [j^2,(j+1)^2], j>k exceeds the number of primes in the interval [k^2,(k+1)^2].

%I A349998 #12 Dec 17 2024 12:57:52
%S A349998 5,9,14,17,23,26,30,42,49,55,56,80,85,89,119,137,143,149,156,174,178,
%T A349998 188,194,200,207,219,228,247,261,263,279,297,327,335,356,425,433,451,
%U A349998 485,506,536,600,607,696,708,749,768,799,801,898,904,955,1015,1059,1110
%N A349998 Numbers k such that the number of primes in any interval [j^2,(j+1)^2], j>k exceeds the number of primes in the interval [k^2,(k+1)^2].
%C A349998 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 A349998 Hugo Pfoertner, <a href="/A349998/b349998.txt">Table of n, a(n) for n = 1..2414</a>
%F A349998 A014085(k) > A014085(a(n)) for all k > a(n).
%e A349998 a(1)=5: There are 2 = A349999(1) primes {29, 31} between 5^2 and 6^2. All intervals between squares above contain at least 3 primes.
%e A349998 a(2)=9: The interval [9^2, 10^2] is the last interval containing not more than 3 = A349999(2) primes {83, 89, 97}.
%e A349998 a(12)=80: The interval [80^2,81^2] is the last interval containing not more than 13 = A349999(12) primes {6421, ..., 6553}.
%e A349998 a(13)=85: The interval [85^2,86^2] is the last interval containing not more than 16 = A349999(13) primes {7229, ..., 7393}.
%Y A349998 Cf. A014085, A349997, A349999.
%K A349998 nonn
%O A349998 1,1
%A A349998 _Hugo Pfoertner_, Dec 09 2021