A228553 - OEIS

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

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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]];