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 A219864 Number of ways to write n as p+q with p and 2pq+1 both prime 26
 0, 0, 1, 1, 2, 3, 0, 2, 4, 2, 2, 4, 1, 2, 6, 3, 1, 2, 2, 5, 3, 1, 1, 7, 2, 6, 3, 1, 6, 8, 2, 2, 5, 3, 3, 8, 2, 4, 6, 3, 4, 4, 1, 3, 7, 2, 3, 7, 3, 6, 8, 2, 1, 12, 5, 4, 7, 4, 7, 7, 7, 5, 4, 4, 6, 9, 2, 2, 13, 2, 5, 7, 2, 4, 18, 6, 3, 5, 6, 5, 8, 4, 2, 9, 4, 10, 5, 2, 5, 17, 3, 3, 7, 7, 5, 8, 3, 3, 17, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Conjecture: a(n)>0 for all n>7. This has been verified for n up to 3*10^8. Zhi-Wei Sun also made the following general conjecture: For each odd integer m not congruent to 5 modulo 6, any sufficiently large integer n can be written as p+q with p and 2*p*q+m both prime. For example, when m = 3, -3, 7, 9, -9, -11, 13, 15, it suffices to require that n is greater than 1, 29, 16, 224, 29, 5, 10, 52 respectively. Sun also guessed that any integer n>4190 can be written as p+q with p, 2*p*q+1, 2*p*q+7 all prime, and any even number n>1558 can be written as p+q with p, q, 2*p*q+3 all prime. He has some other similar observations. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588. EXAMPLE a(10)=2 since 10=3+7=7+3 with 2*3*7+1=43 prime. a(263)=1 since 83 is the only prime p with 2p(263-p)+1 prime. MATHEMATICA a[n_]:=a[n]=Sum[If[PrimeQ[2Prime[k](n-Prime[k])+1]==True, 1, 0], {k, 1, PrimePi[n]}] Do[Print[n, " ", a[n]], {n, 1, 1000}] CROSSREFS Cf. A219842, A002372, A046927, A219838, A219791, A219782, A036468. Sequence in context: A024307 A267852 A328568 * A257844 A194745 A248342 Adjacent sequences:  A219861 A219862 A219863 * A219865 A219866 A219867 KEYWORD nonn AUTHOR Zhi-Wei Sun, Nov 30 2012 STATUS approved

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Last modified August 13 06:22 EDT 2020. Contains 336442 sequences. (Running on oeis4.)