A230224 Number of ways to write 2*n = p + q + r + s with p <= q <= r <= s such that p, q, r, s are primes in A230223.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 4, 1, 3, 3, 3, 4, 4, 3, 5, 4, 5, 3, 6, 4, 6, 5, 5, 5, 7, 5, 9, 4, 6, 6, 8, 6, 9, 5, 7, 7, 10, 6, 8, 7, 8, 7, 9, 5, 10, 7, 11, 7, 7, 7, 11, 7, 10, 6, 10, 6, 13, 7, 9, 7, 11, 9, 11, 7, 9, 6, 14, 8, 12, 6, 13, 11, 12, 11, 13, 10, 16, 9, 14, 7, 14

Offset

1,20

Comments

Conjecture: a(n) > 0 for all n > 17.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Example

a(21) = 1 since 2*21 = 7 + 7 + 11 + 17, and 7, 11, 17 are primes in A230223.
a(27) = 1 since 2*27 = 7 + 11 + 17 + 19, and 7, 11, 17, 19 are primes in A230223.

Mathematica

RQ[n_]:=n>5&&PrimeQ[3n-4]&&PrimeQ[3n-10]&&PrimeQ[3n-14]
SQ[n_]:=PrimeQ[n]&&RQ[n]
a[n_]:=Sum[If[RQ[Prime[i]]&&RQ[Prime[j]]&&RQ[Prime[k]]&&SQ[2n-Prime[i]-Prime[j]-Prime[k]], 1, 0],
{i, 1, PrimePi[n/2]}, {j, i, PrimePi[(2n-Prime[i])/3]}, {k, j, PrimePi[(2n-Prime[i]-Prime[j])/2]}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A002375, A068307, A230223, A230140, A230141, A230219.

Sequence in context: A049099 A181776 A243036 * A206941 A327790 A143179

Adjacent sequences: A230221 A230222 A230223 * A230225 A230226 A230227

Keyword

nonn

Author

Zhi-Wei Sun, Oct 12 2013

Status

approved