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A279727 Sum of the smaller parts of the Goldbach partitions (p,q) of 2n such that all primes from p to q (inclusive) appear as a part in some Goldbach partition of p+q = 2n.

%I A279727 #27 Feb 16 2025 08:33:38
%S A279727 0,0,3,3,8,5,7,0,12,0,11,23,13,0,31,0,17,30,19,0,19,0,23,0,0,23,0,0,
%T A279727 29,101,31,0,0,31,0,0,37,0,37,0,41,109,43,0,43,0,47,0,0,47,0,0,100,0,
%U A279727 0,53,0,0,59,112,61,0,0,61,0,0,67,0,67,0,71,71,73,0,0,73,0,0
%N A279727 Sum of the smaller parts of the Goldbach partitions (p,q) of 2n such that all primes from p to q (inclusive) appear as a part in some Goldbach partition of p+q = 2n.
%H A279727 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H A279727 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H A279727 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A279727 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A279727 a(n) = Sum_{i=3..n} (i * c(i) * c(2n-i) * (Product_{k=i..n} (1-abs(c(k)-c(2n-k))))), where c = A010051.
%p A279727 with(numtheory): A279727:=n->add( i * (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)) * (product(1-abs((pi(k)-pi(k-1))-(pi(2*n-k)-pi(2*n-k-1))), k=i..n)), i=3..n): seq(A279727(n), n=1..100);
%t A279727 Table[Sum[(i Boole[PrimeQ@ i] Boole[PrimeQ[2 n - i]]) Product[1 - Abs[Boole[PrimeQ@ k] - Boole[PrimeQ[2 n - k]]], {k, i, n}], {i, 3, n}], {n, 100}] (* _Michael De Vlieger_, Dec 18 2016 *)
%Y A279727 Cf. A010051, A279315, A279728, A279729.
%K A279727 nonn,easy
%O A279727 1,3
%A A279727 _Wesley Ivan Hurt_, Dec 17 2016