

A230243


Number of primes p < n with 3*p + 8 and (p1)*n + 1 both prime.


1



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
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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.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.


EXAMPLE

a(8) = 1 since 8 = 3 + 5 with 3, 3*3+8 = 17, (31)*8+1 = 17 all prime.
a(17) = 1 since 17 = 7 + 10, and 7, 3*7+8 = 29, (71)*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[n1]}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A034693, A085053, A086685, A023210, A219864, A230217, A230219, A230241.
Sequence in context: A083898 A078314 A068322 * A078687 A133138 A194326
Adjacent sequences: A230240 A230241 A230242 * A230244 A230245 A230246


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Oct 13 2013


STATUS

approved



