login
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.
11
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
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
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Mar 09 2022
STATUS
approved