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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.

%I A352248 #9 Feb 16 2025 08:34:03
%S A352248 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,
%T A352248 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,
%U A352248 10,1,8,6,1,4,2,4,7,1,4,1,3,3,8,2,8,12,2,3,1,3,5
%N 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.
%H A352248 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H A352248 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H A352248 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A352248 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%e A352248 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.
%t A352248 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}];
%t A352248 Table[If[a[n] > 0, a[n], {}], {n, 100}] // Flatten
%Y A352248 Cf. A187797, A278700, A352240, A352283.
%K A352248 nonn
%O A352248 1,5
%A A352248 _Wesley Ivan Hurt_, Mar 09 2022