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

2,3

Comments

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

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

Links

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.)

Formula

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

Example

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

Mathematica

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}]

Crossrefs

Cf. A000040, A000290, A247893, A247895.

Sequence in context: A307553 A327436 A339266 * A278823 A042597 A255635

Adjacent sequences: A247275 A247276 A247277 * A247279 A247280 A247281

Keyword

nonn

Author

Zhi-Wei Sun, Sep 27 2014

Status

approved