A247278 -id:A247278 - cp's OEIS frontend

A247895 Least integer k > 0 such that prime(k) - k*n is prime.

3, 7, 13, 31, 69, 190, 444, 1052, 2702, 6455, 15928, 40073, 100370, 251707, 637321, 1617175, 4124448, 10553415, 27066978, 69709680, 179992909, 465769803, 1208198532, 3140421716, 8179002120, 21338685408, 55762149030, 145935689361, 382465573486, 1003652347100

Offset: 1

Zhi-Wei Sun, Sep 28 2014

nonn

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

(ii) For each integer n > 2, there is a positive integer k with k*n - prime(k) prime.

			a(1) = 3 since prime(3) - 3*1 = 5 - 3 = 2 is prime.

Giovanni Resta, Table of n, a(n) for n = 1..50
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685 [math.NT], 2014-2017.

Cf. A000040, A247278, A247793.

Do[k=1; Label[aa]; If[Prime[k]>k*n&&PrimeQ[Prime[k]-k*n],Print[n," ",k]; Goto[bb]]; k=k+1; Goto[aa]; Label[bb]; Continue,{n,1,22}]

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

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

A247895 Least integer k > 0 such that prime(k) - k*n is prime.

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