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 A187757 Number of ways to write n=x+y (x,y>0) with 6x-1, 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 m-2 contains no prime for any integer m>1. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..20000 Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588. EXAMPLE a(92)=1 since 92=40+52 with 6*40-1, 6*40+1, 6*52+1 and 6*52+5 all prime. MATHEMATICA a[n_]:=a[n]=Sum[If[PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True&&PrimeQ[6(n-k)+1]==True&&PrimeQ[6(n-k)+5]==True, 1, 0], {k, 1, n-1}] 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 Zhi-Wei Sun, Jan 03 2013 STATUS approved

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Last modified September 25 20:01 EDT 2021. Contains 347659 sequences. (Running on oeis4.)