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

1,2

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.

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

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A301254 A291318 A178379 * A356928 A357807 A337568

Adjacent sequences: A228550 A228551 A228552 * A228554 A228555 A228556

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 25 2013

Status

approved