

A219842


Number of ways to write n as x+y (0<x<=y) with 2x*y+1 prime.


20



0, 1, 1, 1, 1, 3, 1, 1, 4, 2, 2, 4, 2, 2, 7, 4, 1, 3, 4, 5, 5, 4, 2, 8, 2, 7, 4, 2, 8, 11, 5, 3, 8, 7, 5, 14, 7, 5, 10, 8, 7, 8, 4, 8, 9, 5, 4, 11, 6, 11, 14, 5, 3, 19, 12, 7, 11, 6, 9, 12, 13, 8, 9, 10, 12, 16, 5, 6, 22, 8, 11, 11, 5, 10, 26, 15, 5, 11, 15, 10
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OFFSET

1,6


COMMENTS

Conjecture: a(n)>0 for all n>1. Moreover, any integer n>357 can be written as x+y (x>0, y>0) with 2x*y+1 and 2x*y1 twin primes.
This conjecture has been verified for n up to 10^8, and it implies the twin prime conjecture.
ZhiWei Sun also made the following general conjecture: For each positive odd integer m, any sufficiently large integer n can be written as x+y, where x and y are positive integers with 2x*y+m and 2x*ym both prime.
For example, when m=3,5,7,9,11 it suffices to require that n is greater than 5090, 222, 1785, 548, 603 respectively.


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=1+9=3+7 with 2*1*9+1=19 and 2*3*7+1=43 both prime.


MATHEMATICA

a[n_] := a[n] = Sum[If[PrimeQ[2k(nk)+1] == True, 1, 0], {k, n/2}]; Do[Print[n, " ", a[n]], {n, 100}]


CROSSREFS

Cf. A091182, A219838, A219782.
Sequence in context: A188139 A134557 A290342 * A134264 A125181 A157076
Adjacent sequences: A219839 A219840 A219841 * A219843 A219844 A219845


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Nov 29 2012


STATUS

approved



