A230243 Number of primes p < n with 3*p + 8 and (p-1)*n + 1 both prime.

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 4, 2, 1, 4, 2, 2, 4, 2, 3, 2, 4, 3, 4, 4, 2, 2, 2, 1, 5, 3, 4, 3, 3, 2, 3, 4, 2, 2, 4, 2, 4, 4, 1, 5, 3, 2, 6, 4, 1, 5, 6, 3, 3, 5, 1, 5, 5, 2, 7, 5, 3, 4, 4, 3, 4, 6, 3, 4, 6, 4, 5, 6, 3, 7, 4, 2, 6, 1, 3, 5, 9, 3, 3, 7, 4, 3, 7, 1, 6, 5, 5, 5, 6, 3, 6, 7

Offset

1,9

Comments

Conjecture: a(n) > 0 for all n > 4.

This implies A. Murthy's conjecture (cf. A034693) that for any integer n > 1, there is a positive integer k < n such that k*n + 1 is prime.

Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 21 2023

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Example

a(8) = 1 since 8 = 3 + 5 with 3, 3*3+8 = 17, (3-1)*8+1 = 17 all prime.
a(17) = 1 since 17 = 7 + 10, and 7, 3*7+8 = 29, (7-1)*17+1 = 103 are all prime.

Mathematica

a[n_]:=Sum[If[PrimeQ[3Prime[i]+8]&&PrimeQ[(Prime[i]-1)n+1], 1, 0], {i, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A034693, A085053, A086685, A023210, A219864, A230217, A230219, A230241.

Sequence in context: A330894 A068322 A340967 * A338898 A078687 A133138

Adjacent sequences: A230240 A230241 A230242 * A230244 A230245 A230246

Keyword

nonn

Author

Zhi-Wei Sun, Oct 13 2013

Status

approved