login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A376917
Starting from Goldbach decomposition of 10 = p + q = 5 + 5, 12 = 7 + 5, and 14 = 7 + 7, a(n) is the first number in A001057 such that if 2n - 6 = p + q, 2n = p' + q', where p' = p + 6 * a(n) and q' = 2n - q' are both primes.
0
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, -1, 1, 1, 0, 1, -1, 0, 1, 0, 1, -1, 0, 2, 0, 0, 0, 0, 1, 1, 0, -1, 1, 1, 1, -2, -1, -1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 1, 1, 0, 1, -1, 0, -2, 0, 1, 0, 0, 2, 0, 1, 0, 1, 2, 1, 1, 0, 1, -2, 1, -2, 1, 1, 0, 1, 1, 1, -1, 2, 1, 1
OFFSET
8,27
COMMENTS
By definition, this sequence starts from n = 8.
Hypothesis: a(n) is defined for all n >= 8 and for all n >=8, the corresponding Goldbach decomposition 2n = p + q has positive primes p and q.
EXAMPLE
When n = 8, 2n = 16. 2n - 6 = 10. 10 = p + q = 5 + 5 (by definition). a(8) = 0, p' = p + a(8) = 5, q' = 2n - p' = 16 - 5 = 11. P' and q' are both primes.
...
When n = 10, 2n = 20. 2n - 6 = 14. 14 = p + q = 7 + 7 (by definition). a(10) = 0, p' = p + a(10) = 7, q' = 2n - p' = 20 - 7 = 13. P' and q' are both primes.
...
When n = 13, 2n = 26. 2n - 6 = 20. 20 = p + q = 7 + 13 (per above evaluation). a(13) = 0, p' = p + a(13) = 7, q' = 2n - p' = 26 - 7 = 19. P' and q' are both primes.
When n = 16, 2n = 32. 2n - 6 = 26. 26 = p + q = 7 + 19 (per above evaluation). a(16) = 1, p' = p + a(16) = 13, q' = 2n - p' = 32 - 13 = 19. P' and q' are both primes. It is tested when a(16) is 0, q' = 25 is not a prime, thus a(16) = 1 is the first number in A001057 that makes both p' and q' primes.
MATHEMATICA
a = {}; p = {5, 7, 7}; Do[Do[n = 6*k - 4 + 2*j; i = 0; While[i++; m = 1/4 + (i - 1/2)*(-1)^i/2; pr = p[[j]] + 6*m; q = n - pr; ! (PrimeQ[pr] && PrimeQ[q])]; p[[j]] = pr; AppendTo[a, m], {j, 1, 3}], {k, 3, 30}]; Print[a]
CROSSREFS
Ref. A001057; Cf. A045917
Sequence in context: A307666 A319995 A266344 * A334944 A174875 A193510
KEYWORD
sign,easy
AUTHOR
Lei Zhou, Oct 09 2024
STATUS
approved