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 A230219 Number of ways to write 2*n + 1 = p + q + r with p <= q such that p, q, r are primes in A230217 and p + q + 9 is also prime. 14
 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 5, 2, 2, 4, 3, 1, 4, 3, 1, 4, 1, 2, 5, 2, 3, 4, 3, 3, 8, 6, 3, 12, 6, 2, 13, 3, 3, 7, 6, 4, 5, 4, 4, 8, 7, 4, 12, 7, 3, 19, 6, 3, 16, 5, 4, 9, 5, 5, 7, 10, 4, 5, 8, 3, 14, 4, 3, 14, 2, 5, 12, 5, 2, 14, 9, 2, 10, 12, 4, 12, 7, 6, 12, 7, 9, 14, 8, 6, 12, 5, 4, 19, 8, 4, 23, 6, 3, 14 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,10 COMMENTS Conjecture: a(n) > 0 for all n > 6. This implies Goldbach's weak conjecture for odd numbers and also Goldbach's conjecture for even numbers. The conjecture also implies that there are infinitely many primes in A230217. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588. EXAMPLE a(18) = 1 since 2*18 + 1 = 7 + 13 + 17, and 7, 13, 17 are terms of A230217, and 7 + 13 + 9 = 29 is a prime. MATHEMATICA RQ[n_]:=PrimeQ[n+6]&&PrimeQ[3n+8] SQ[n_]:=PrimeQ[n]&&RQ[n] a[n_]:=Sum[If[RQ[Prime[i]]&&RQ[Prime[j]]&&PrimeQ[Prime[i]+Prime[j]+9]&&SQ[2n+1-Prime[i]-Prime[j]], 1, 0], {i, 1, PrimePi[n-1]}, {j, i, PrimePi[2n-2-Prime[i]]}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A002375, A068307, A230217, A230140, A230141. Sequence in context: A165195 A121487 A057031 * A147292 A078391 A153206 Adjacent sequences:  A230216 A230217 A230218 * A230220 A230221 A230222 KEYWORD nonn AUTHOR Zhi-Wei Sun, Oct 11 2013 STATUS approved

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Last modified August 18 10:34 EDT 2022. Contains 356212 sequences. (Running on oeis4.)