

A219864


Number of ways to write n as p+q with p and 2pq+1 both prime


26



0, 0, 1, 1, 2, 3, 0, 2, 4, 2, 2, 4, 1, 2, 6, 3, 1, 2, 2, 5, 3, 1, 1, 7, 2, 6, 3, 1, 6, 8, 2, 2, 5, 3, 3, 8, 2, 4, 6, 3, 4, 4, 1, 3, 7, 2, 3, 7, 3, 6, 8, 2, 1, 12, 5, 4, 7, 4, 7, 7, 7, 5, 4, 4, 6, 9, 2, 2, 13, 2, 5, 7, 2, 4, 18, 6, 3, 5, 6, 5, 8, 4, 2, 9, 4, 10, 5, 2, 5, 17, 3, 3, 7, 7, 5, 8, 3, 3, 17, 8
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OFFSET

1,5


COMMENTS

Conjecture: a(n)>0 for all n>7.
This has been verified for n up to 3*10^8.
ZhiWei Sun also made the following general conjecture: For each odd integer m not congruent to 5 modulo 6, any sufficiently large integer n can be written as p+q with p and 2*p*q+m both prime.
For example, when m = 3, 3, 7, 9, 9, 11, 13, 15, it suffices to require that n is greater than 1, 29, 16, 224, 29, 5, 10, 52 respectively.
Sun also guessed that any integer n>4190 can be written as p+q with p, 2*p*q+1, 2*p*q+7 all prime, and any even number n>1558 can be written as p+q with p, q, 2*p*q+3 all prime. He has some other similar observations.


LINKS

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


EXAMPLE

a(10)=2 since 10=3+7=7+3 with 2*3*7+1=43 prime.
a(263)=1 since 83 is the only prime p with 2p(263p)+1 prime.


MATHEMATICA

a[n_]:=a[n]=Sum[If[PrimeQ[2Prime[k](nPrime[k])+1]==True, 1, 0], {k, 1, PrimePi[n]}]
Do[Print[n, " ", a[n]], {n, 1, 1000}]


CROSSREFS

Cf. A219842, A002372, A046927, A219838, A219791, A219782, A036468.
Sequence in context: A024307 A267852 A328568 * A257844 A194745 A248342
Adjacent sequences: A219861 A219862 A219863 * A219865 A219866 A219867


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Nov 30 2012


STATUS

approved



