A163407 Sum of semiprime divisors of n with repetition.

0, 0, 0, 4, 0, 6, 0, 12, 9, 10, 0, 16, 0, 14, 15, 24, 0, 21, 0, 24, 21, 22, 0, 30, 25, 26, 27, 32, 0, 31, 0, 40, 33, 34, 35, 37, 0, 38, 39, 42, 0, 41, 0, 48, 39, 46, 0, 48, 49, 45, 51, 56, 0, 45, 55, 54, 57, 58, 0, 51, 0, 62, 51, 60, 65, 61, 0, 72, 69, 59, 0, 57, 0, 74, 55, 80, 77, 71, 0

Offset

1,4

Comments

We regard each prime divisor of n as distinct, and count each product of an unordered, distinct pair of them as a semiprime divisor.

Links

Table of n, a(n) for n=1..79.

Formula

If s is the sum of the prime divisors of n with repetition, and ss is the sum of their squares, a(n) = (s^2 - ss) / 2.

Example

For n = 12, the prime divisors with repetition are 2,2,3. Distinguishing the 2s as 2 and 2', we have semiprime divisors 2*2', 2*3, and 2'*3, totaling 4+6+6 = 16.

Prog

(PARI) a(n)=local(fn, p, e, s, ss); fn=factor(n); for(i=1, matsize(fn)[1], p=fn[i, 1]; e=fn[i, 2]; s+=p*e; ss+=p^2*e); (s^2-ss)\2

Crossrefs

Cf. A076290, A001358, A001222, A067666.

Sequence in context: A191558 A075083 A179939 * A023891 A075091 A132953

Adjacent sequences: A163404 A163405 A163406 * A163408 A163409 A163410

Keyword

nonn

Author

Franklin T. Adams-Watters, Jul 26 2009

Status

approved