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 A219157 Number of prime pairs {p,q} with p>q and p-2,q+2 also prime such that p+(1+mod(-n,6))q=n if n is not congruent to 2 mod 6, and p-q=n and q
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 0, 2, 0, 2, 2, 1, 1, 2, 3, 1, 0, 2, 1, 1, 0, 2, 2, 1, 2, 1, 2, 1, 0, 1, 0, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 0, 1, 3, 1, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,22 COMMENTS Conjecture: a(n)>0 for all n>30000 with n different from 38451, 46441, 50671, 62371. This conjecture is stronger than the twin prime conjecture. It is similar to the conjecture associated with A219055 about sexy prime pairs. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..100000 Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588. EXAMPLE a(16)=1 since 16=7+3*3 with 7-2 and 3+2 prime. a(26)=1 since 26=31-5 with 31-2 and 5+2 prime. MATHEMATICA c[n_]:=c[n]=If[Mod[n-2, 6]==0, 1, -1-Mod[-n, 6]] d[n_]:=d[n]=2+If[Mod[n-2, 6]>0, Mod[-n, 6], 0] a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+2]==True&&PrimeQ[n+c[n]Prime[k]]==True&&PrimeQ[n+c[n]Prime[k]-2]==True, 1, 0], {k, 1, PrimePi[(n-1)/d[n]]}] Do[Print[n, " ", a[n]], {n, 1, 100000}] CROSSREFS Cf. A001359, A006512, A002375, A046927, A219055, A218754, A218825, A219052. Sequence in context: A194636 A286299 A081729 * A080215 A266871 A331290 Adjacent sequences:  A219154 A219155 A219156 * A219158 A219159 A219160 KEYWORD nonn AUTHOR Zhi-Wei Sun, Nov 12 2012 STATUS approved

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Last modified January 28 00:30 EST 2020. Contains 331301 sequences. (Running on oeis4.)