cp's OEIS Frontend

A247278 Least integer k > 0 such that k*n - prime(k) is a square.

%I A247278 #38 Apr 23 2020 12:11:58
%S A247278 1,1,4,29,1,3,4,43,3,1,5,37,2,5,9,19,1,267,22,23,4,3,43,57,2,1,46,19,
%T A247278 20,5,4,23,440,3,5,162,1,7,20,499,2,74,4,128,29,9,927,215,156,1,96,91,
%U A247278 7,1058,73,162,3,763,5
%N A247278 Least integer k > 0 such that k*n - prime(k) is a square.
%C A247278 Conjecture: a(n) exists for any n > 1.
%C A247278 Note that k*n - prime(k) < 0 if k > e^(n + 1).
%H A247278 Zhi-Wei Sun, <a href="/A247278/b247278.txt">Table of n, a(n) for n = 2..10000</a>
%H A247278 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1409.5685">A new theorem on the prime-counting function</a>, arXiv:1409.5685, 2014.
%H A247278 Zhi-Wei Sun, <a href="http://dx.doi.org/doi:10.1007/s11139-015-9702-z">A new theorem on the prime-counting function</a>, Ramanujan J. 42 (2017), no.1, 59-67. (Cf. Conjecture 4.1.)
%F A247278 a(A059100(n)) = 1. - _Michel Marcus_, Sep 28 2014
%e A247278 a(5) = 29 since 29 * 5 - prime(29) = 145 - 109 = 6^2.
%t A247278 SQ[n_] := IntegerQ[Sqrt[n]]
%t A247278 Do[k = 1; Label[aa]; If[SQ[k * n - Prime[k]], Print[n, " ", k]; Goto[bb]]; k = k + 1; Goto[aa]; Label[bb]; Continue,{n, 2, 60}]
%Y A247278 Cf. A000040, A000290, A247893, A247895.
%K A247278 nonn
%O A247278 2,3
%A A247278 _Zhi-Wei Sun_, Sep 27 2014