

A187757


Number of ways to write n=x+y (x,y>0) with 6x1, 6x+1, 6y+1 and 6y+5 all prime


6



0, 1, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, 4, 4, 2, 3, 2, 6, 6, 5, 4, 2, 6, 5, 4, 4, 2, 6, 4, 4, 4, 3, 5, 7, 5, 5, 3, 4, 9, 5, 6, 4, 5, 6, 4, 5, 5, 6, 7, 6, 6, 3, 7, 7, 6, 6, 4, 6, 6, 5, 6, 4, 7, 6, 7, 2, 3, 7, 7, 7, 5, 3, 5, 5, 7, 8, 5, 8, 8, 4, 5, 4, 10, 10, 6, 6, 2, 9, 6, 9, 7, 1, 8, 4, 5, 7, 3, 9, 5, 3
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

Conjecture: a(n)>0 for all n>1.
This has been verified for n up to 10^9. It implies that there are infinitely many twin primes and also infinitely many cousin primes, since the interval [m!+2,m!+m] of length m2 contains no prime for any integer m>1.


LINKS

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


EXAMPLE

a(92)=1 since 92=40+52 with 6*401, 6*40+1, 6*52+1 and 6*52+5 all prime.


MATHEMATICA

a[n_]:=a[n]=Sum[If[PrimeQ[6k1]==True&&PrimeQ[6k+1]==True&&PrimeQ[6(nk)+1]==True&&PrimeQ[6(nk)+5]==True, 1, 0], {k, 1, n1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]


CROSSREFS

Cf. A001359, A006512, A023200, A046132, A219157, A218867, A219185, A220455.
Sequence in context: A205717 A304689 A288677 * A286529 A306225 A077199
Adjacent sequences: A187754 A187755 A187756 * A187758 A187759 A187760


KEYWORD

nonn,nice


AUTHOR

ZhiWei Sun, Jan 03 2013


STATUS

approved



