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.
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, 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, 0, 53, 0, 0, 59, 112, 61, 0, 0, 61, 0, 0, 67, 0, 67, 0, 71, 71, 73, 0, 0, 73, 0, 0
Offset: 1
Links
- Eric Weisstein's World of Mathematics, Goldbach Partition
- Wikipedia, Goldbach's conjecture
- Index entries for sequences related to Goldbach conjecture
- Index entries for sequences related to partitions
Programs
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Maple
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);
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Mathematica
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 *)
Formula
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.