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 A220431 Number of ways to write n=x+y (x>0, y>0) with 3x-1, 3x+1 and xy-1 all prime. 5
 0, 0, 0, 1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 6, 1, 3, 6, 4, 3, 3, 2, 3, 4, 3, 4, 2, 3, 3, 5, 4, 4, 7, 1, 2, 5, 1, 5, 7, 4, 2, 3, 7, 4, 7, 2, 4, 7, 4, 4, 5, 2, 5, 8, 4, 3, 3, 5, 2, 8, 5, 4, 3, 10, 7, 8, 2, 3, 5, 5, 3, 6, 3, 3, 14, 4, 3, 12, 3, 7, 7, 5, 6, 8, 7, 5, 9, 9, 4, 4, 3, 6, 10, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Conjecture: a(n)>0 for all n>3. This has been verified for n up to 10^8, and it is stronger than A. Murthy's conjecture related to A109909. Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 26 2023 The conjecture implies the twin prime conjecture for the following reason: If x_1<...2 not equal to 63 can be written as x+y (x>0, y>0) with 2x-1, 2x+1 and 2xy+1 all prime. Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 26 2023 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(22)=1 since 22=4+18 with 3*4-1, 3*4+1 and 4*18-1 all prime. MATHEMATICA a[n_]:=a[n]=Sum[If[PrimeQ[3k-1]==True&&PrimeQ[3k+1]==True&&PrimeQ[k(n-k)-1]==True, 1, 0], {k, 1, n-1}] Do[Print[n, " ", a[n]], {n, 1, 1000}] CROSSREFS Cf. A001359, A006512, A220419, A220413, A173587, A220272, A219842, A219864, A219923. Sequence in context: A305976 A074592 A089993 * A351284 A268317 A234092 Adjacent sequences: A220428 A220429 A220430 * A220432 A220433 A220434 KEYWORD nonn AUTHOR Zhi-Wei Sun, Dec 14 2012 STATUS approved

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Last modified September 23 14:27 EDT 2023. Contains 365551 sequences. (Running on oeis4.)