A228553 - OEIS

%I #31 Feb 02 2023 15:28:24

%S 0,4,9,15,46,35,82,94,142,142,263,357,371,302,591,334,780,980,578,821,

%T 1340,785,1356,1987,1512,1353,2677,1421,2320,4242,1955,2803,4362,1574,

%U 4021,5298,4177,4159,6731,4132,5593,9808

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

%C 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.

%H Robert Israel, <a href="/A228553/b228553.txt">Table of n, a(n) for n = 1..10000</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=2..n} c(i) * c(2*n-i) * i * (2*n-i), where c = A010051.

%F 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

%e 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.

%p 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);

%p # Alternative:

%p f:= proc(n)

%p   local S;

%p   S:= select(t -> isprime(t) and isprime(2*n-t), [seq(i,i=3..n,2)]);

%p   add(t*(2*n-t),t=S)

%p end proc:

%p f(2):= 4:

%p map(f, [$1..200]); # _Robert Israel_, Nov 29 2020

%t c[n_] := Boole[PrimeQ[n]];

%t a[n_] := Sum[c[i]*c[2n-i]*i*(2n-i), {i, 2, n}];

%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Feb 02 2023 *)

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

%K nonn

%O 1,2

%A _Wesley Ivan Hurt_, Aug 25 2013