A247895 -id:A247895 - cp's OEIS frontend

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

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, 20, 5, 4, 23, 440, 3, 5, 162, 1, 7, 20, 499, 2, 74, 4, 128, 29, 9, 927, 215, 156, 1, 96, 91, 7, 1058, 73, 162, 3, 763, 5

Offset: 2

Zhi-Wei Sun, Sep 27 2014

nonn

Conjecture: a(n) exists for any n > 1.

Note that k*n - prime(k) < 0 if k > e^(n + 1).

			a(5) = 29 since 29 * 5 - prime(29) = 145 - 109 = 6^2.

Zhi-Wei Sun, Table of n, a(n) for n = 2..10000
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.
Zhi-Wei Sun, A new theorem on the prime-counting function, Ramanujan J. 42 (2017), no.1, 59-67. (Cf. Conjecture 4.1.)

Cf. A000040, A000290, A247893, A247895.

SQ[n_] := IntegerQ[Sqrt[n]]
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}]

a(A059100(n)) = 1. - Michel Marcus, Sep 28 2014

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

Original entry on oeis.org

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

Zhi-Wei Sun, Sep 27 2014

nonn

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

See also A247278 for a related conjecture.

			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.

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.)

Cf. A000040, A000290, A247278, A247895.

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 *)

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

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

cp's OEIS Frontend

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

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