A269801 Total sum of the divisors of the primes p,q such that n=p+q and p>=q.

0, 0, 0, 0, 6, 7, 8, 9, 10, 11, 24, 0, 14, 15, 32, 17, 36, 0, 40, 21, 44, 23, 72, 0, 78, 27, 84, 0, 60, 0, 96, 33, 68, 35, 144, 0, 152, 0, 80, 41, 126, 0, 176, 45, 138, 47, 192, 0, 250, 51, 208, 0, 162, 0, 280, 57, 174, 0, 240, 0, 372, 63, 192, 65, 330, 0

Offset: 0

Author

Wesley Ivan Hurt, Mar 05 2016

Keywords

• nonn
• easy

Examples

			a(5) = 7; Since 5 can be expressed in one way as the sum of the two primes 2 and 3, we add the sum of their divisors separately: sigma(2) + sigma(3) = 3 + 4 = 7.
a(10) = 24; Since 10 can be expressed in two ways as the sum of two primes, we add the sum of the divisors of each prime p and q: 10 = 3+7 = 5+5, so sigma(3) + sigma(7) + sigma(5) + sigma(5) = 4 + 8 + 6 + 6 = 24.
		

Crossrefs

Cf. A000203 (sigma), A010051, A014092, A061358.

Programs

Maple

with(numtheory): A269801:=n->(n+2)*sum((pi(i)-pi(i-1))*(pi(n-i)-pi(n-i-1)), i=2..floor(n/2)): seq(A269801(n), n=0..100);

Mathematica

Table[(n+2) Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]), {i, 2, Floor[n/2]}], {n, 0, 80}]

PARI

a(n) = sum(i=0, n\2, if (isprime(i) && isprime(n-i), sigma(i)+sigma(n-i))); \\ Michel Marcus, Mar 05 2016

Formula

a(n) = (n+2) * A061358(n).

a(n) = (n+2) * Sum_{i=2..floor(n/2)} A010051(i) * A010051(n-i).

a(n) = Sum_{i=2..floor(n/2)} (A000203(i) + A000203(n-i)) * A010051(i) * A010051(n-i).