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A279729 Sum of all the 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 A279729 #14 Feb 16 2025 08:33:38
%S A279729 0,0,6,8,20,12,14,0,36,0,22,72,26,0,90,0,34,72,38,0,42,0,46,0,0,52,0,
%T A279729 0,58,300,62,0,0,68,0,0,74,0,78,0,82,252,86,0,90,0,94,0,0,100,0,0,212,
%U A279729 0,0,112,0,0,118,240,122,0,0,128,0,0,134,0,138,0,142,144,146,0
%N A279729 Sum of all the 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 A279729 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H A279729 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H A279729 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A279729 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A279729 a(n) = 2n * A278700(n).
%F A279729 a(n) = A279727(n) + A279728(n).
%p A279729 with(numtheory): A279729:=n->2*n*add((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(A279729(n), n=1..100);
%t A279729 f[n_, x_: 0] := Sum[(If[x == 0, i, 2 n - 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}]; Table[f@ n + f[n, 1], {n, 100}] (* _Michael De Vlieger_, Dec 18 2016 *)
%Y A279729 Cf. A278700, A279727, A279728.
%K A279729 nonn,easy
%O A279729 1,3
%A A279729 _Wesley Ivan Hurt_, Dec 17 2016