A352248 Number of pairs of Goldbach partitions of A352240(n), (p,q) and (r,s) with p,q,r,s prime and p < r <= s < q, such that all integers in the open intervals (p,r) and (s,q) are composite.

1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 3, 4, 1, 2, 2, 2, 3, 1, 4, 6, 1, 1, 4, 2, 3, 1, 2, 7, 8, 5, 4, 1, 3, 1, 2, 5, 7, 1, 3, 1, 3, 6, 4, 7, 2, 4, 1, 1, 3, 1, 2, 5, 2, 7, 14, 4, 1, 2, 3, 1, 2, 2, 1, 2, 7, 1, 10, 1, 8, 6, 1, 4, 2, 4, 7, 1, 4, 1, 3, 3, 8, 2, 8, 12, 2, 3, 1, 3, 5

Offset

1,5

Links

Table of n, a(n) for n=1..90.

Eric Weisstein's World of Mathematics, Goldbach Partition

Wikipedia, Goldbach's conjecture

Index entries for sequences related to Goldbach conjecture

Index entries for sequences related to partitions

Example

a(13) = 4; The Goldbach partitions of A352240(13) = 60 are: 7+53 = 13+47 = 17+43 = 19+41 = 23+37 = 29+31. The 4 pairs of Goldbach partitions of 60 that are being counted are: (13,47),(17,43); (17,43),(19,41); (19,41),(23,37); and (23,37),(29,31). Note that the pair (7,53),(13,47) is not counted since there is a prime in the interval (7,13), namely 11.

Mathematica

a[n_] := Sum[Sum[KroneckerDelta[NextPrime[k], i]*KroneckerDelta[NextPrime[2 n - i], 2 n - k]*(PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - k] - PrimePi[2 n - k - 1]) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {k, i}], {i, n}];
Table[If[a[n] > 0, a[n], {}], {n, 100}] // Flatten

Crossrefs

Cf. A187797, A278700, A352240, A352283.

Sequence in context: A369174 A355855 A046816 * A301475 A138328 A137264

Adjacent sequences: A352245 A352246 A352247 * A352249 A352250 A352251

Keyword

nonn

Author

Wesley Ivan Hurt, Mar 09 2022

Status

approved