A228553 Sum of the products formed by multiplying together the smaller and larger parts of each Goldbach partition of 2n.

0, 4, 9, 15, 46, 35, 82, 94, 142, 142, 263, 357, 371, 302, 591, 334, 780, 980, 578, 821, 1340, 785, 1356, 1987, 1512, 1353, 2677, 1421, 2320, 4242, 1955, 2803, 4362, 1574, 4021, 5298, 4177, 4159, 6731, 4132, 5593, 9808

Offset: 1

Author

Wesley Ivan Hurt, Aug 25 2013

Keywords

• nonn

Comments

Since the product of each prime pair is semiprime and since we are adding A045917(n) of these, a(n) is expressible as the sum of exactly A045917(n) distinct semiprimes.

Examples

			a(5) = 46. 2*5 = 10 has two Goldbach partitions: (7,3) and (5,5). Taking the products of the larger and smaller parts of these partitions and adding, we get 7*3 + 5*5 = 46.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to Goldbach conjecture
• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A045917, A064911, A105020, A185297, A187129.

Programs

Maple

with(numtheory); seq(sum( (2*k*i-i^2) * (pi(i)-pi(i-1)) * (pi(2*k-i)-pi(2*k-i-1)),  i=2..k), k=1..70);
# Alternative:
f:= proc(n)
  local S;
  S:= select(t -> isprime(t) and isprime(2*n-t), [seq(i,i=3..n,2)]);
  add(t*(2*n-t),t=S)
end proc:
f(2):= 4:
map(f, [$1..200]); # Robert Israel, Nov 29 2020

Mathematica

c[n_] := Boole[PrimeQ[n]];
a[n_] := Sum[c[i]*c[2n-i]*i*(2n-i), {i, 2, n}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 02 2023 *)

Formula

a(n) = Sum_{i=2..n} c(i) * c(2*n-i) * i * (2*n-i), where c = A010051.

a(n) = Sum_{k=(n^2-n+2)/2..(n^2+n-2)/2} c(A105020(k)) * A105020(k), where c = A064911. - Wesley Ivan Hurt, Sep 19 2021