A247893 Least integer k > 0 such that prime(k) - k*n is a square.

1, 1, 12, 35, 75, 181, 490, 1061, 2707, 6459, 15932, 40127, 100362, 251711, 637236, 1617181, 4124444, 10553419, 27066987, 69709706, 179992917, 465769804, 1208198534, 3140421726, 8179002096, 21338685437, 55762149044, 145935689364, 382465573484, 1003652347334

Offset: 1

Author

Zhi-Wei Sun, Sep 27 2014

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0.

See also A247278 for a related conjecture.

Examples

			a(3) = 12 with prime(12) - 12*3 = 37 - 36 = 1^2.
a(21) = 179992917 with prime(179992917) - 179992917*21 = 3779851261 - 179992917*21 = 2^2.
a(22) = 465769804 with prime(465769804) - 465769804*22 = 10246935737 - 465769804*22 = 7^2.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..50
• Zhi-Wei Sun, A new theorem on the prime-counting function, Ramanujan J. 42 (2017), no.1, 59-67. (Cf. Conjecture 4.1.)

Crossrefs

Cf. A000040, A000290, A247278, A247895.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[k=1;Label[aa];If[SQ[Prime[k]-k*n],Print[n," ",k];Goto[bb]];k=k+1;Goto[aa];Label[bb];Continue,{n,1,18}]
lik[n_]:=Module[{k=1},While[!IntegerQ[Sqrt[Prime[k]-k*n]],k++];k]; Array[lik,20] (* Harvey P. Dale, May 11 2019 *)

Extensions

a(21)-a(22) from Zhi-Wei Sun, Apr 21 2020

Terms a(23) and beyond from Giovanni Resta, Apr 22 2020