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

Author

Zhi-Wei Sun, Sep 28 2014

Keywords

• nonn

Comments

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.

Examples

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

Links

• 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.

Crossrefs

Cf. A000040, A247278, A247793.

Programs

Mathematica

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}]

Extensions

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