

A219026


Number of primes p<=n such that 2np and 2n+p2 are both prime


3



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

1,5


COMMENTS

Conjecture: a(n)>0 except for n=1,2,4,6,10,22,57.
This is stronger than the Goldbach conjecture; it has been verified for n up to 5*10^7.
ZhiWei Sun also conjectured that if n is not among 1,2,3,5,8,87,108 then there is a prime p in (n,2n)
such that 2np and 2n+p2 are both prime. For conjectures in Section 2 of arXiv:1211.1588, he had similar conjectures with p<=n replaced by p in (n,2n)
For example, if n is not among 1,2,4,6,10,15 then there is a prime p in (n,2n) such that
2np and 2n+p+2 are both prime.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..20000.
ZhiWei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588v2.


EXAMPLE

a(8)=2 since 3 and 5 are the only primes p<=8 with 16p and 14+p both prime.


MATHEMATICA

a[n_]:=a[n]=Sum[If[PrimeQ[2nPrime[k]]==True&&PrimeQ[2n+Prime[k]2]==True, 1, 0], {k, 1, PrimePi[n]}]
Do[Print[n, " ", a[n]], {n, 1, 20000}]
np[n_]:=Count[Prime[Range[PrimePi[n]]], _?(AllTrue[{2n#, 2n+#2}, PrimeQ]&)]; Array[np, 100] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 23 2017 *)


CROSSREFS

Cf. A000040, A002375, A218754, A218585, A218654, A218656, A218825, A219023, A219025.
Sequence in context: A036586 A092928 A321090 * A085097 A117997 A079684
Adjacent sequences: A219023 A219024 A219025 * A219027 A219028 A219029


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Nov 10 2012


STATUS

approved



