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A352240 Even numbers with at least one pair of Goldbach partitions, (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 A352240 #26 Feb 16 2025 08:34:03
%S A352240 10,16,18,22,24,30,34,36,42,46,48,54,60,64,66,72,76,78,82,84,90,96,98,
%T A352240 102,106,108,110,112,114,120,126,132,136,138,140,142,144,150,154,156,
%U A352240 160,162,168,174,180,184,186,188,190,192,194,196,198,202,204,210,216,218,220,222
%N A352240 Even numbers with at least one pair of Goldbach partitions, (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.
%C A352240 Similar to A187797 but also contains the numbers 82, 96, 98, 110, 136, ...
%H A352240 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H A352240 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H A352240 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A352240 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A352240 a(n) = A352442(n) + A352443(n).
%F A352240 a(n) = A352444(n) + A352445(n).
%e A352240 82 is in the sequence since it has at least one pair of Goldbach partitions, namely (23,59) and (29,53), such that all integers in the open intervals (23,29) and (53,59) are composite.
%t A352240 Table[If[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}] > 0, 2 n, {}], {n, 150}] // Flatten
%Y A352240 Cf. A187797, A278700, A352248, A352283.
%Y A352240 Cf. A352442, A352443, A352444, A352445.
%K A352240 nonn
%O A352240 1,1
%A A352240 _Wesley Ivan Hurt_, Mar 08 2022