A279728 - OEIS

%I #22 Nov 19 2021 15:45:18

%S 0,0,3,5,12,7,7,0,24,0,11,49,13,0,59,0,17,42,19,0,23,0,23,0,0,29,0,0,

%T 29,199,31,0,0,37,0,0,37,0,41,0,41,143,43,0,47,0,47,0,0,53,0,0,112,0,

%U 0,59,0,0,59,128,61,0,0,67,0,0,67,0,71,0,71,73,73,0,0,79,0,0

%N Sum of the larger 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = Sum_{i=3..n} ((2n-i) * c(i) * c(2n-i) * (Product_{k=i..n} (1-abs(c(k) - c(2n-k))))), where c = A010051.

%p with(numtheory): A279728:=n->add( (2*n-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(A279728(n), n=1..100);

%t Table[Sum[((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}], {n, 100}] (* _Michael De Vlieger_, Dec 18 2016 *)

%Y Cf. A010051, A279315, A279727, A279729.

%K nonn,easy

%O 1,3

%A _Wesley Ivan Hurt_, Dec 17 2016