A339436 If n = p_1 * ... * p_m with primes p_i <= p_{i+1}, a(n) = Sum_{j=1..m-1} p_1*...*p_j + Sum_{j=2..m} p_j*...*p_m.

0, 0, 4, 0, 5, 0, 12, 6, 7, 0, 15, 0, 9, 8, 28, 0, 20, 0, 21, 10, 13, 0, 35, 10, 15, 24, 27, 0, 28, 0, 60, 14, 19, 12, 48, 0, 21, 16, 49, 0, 36, 0, 39, 32, 25, 0, 75, 14, 42, 20, 45, 0, 65, 16, 63, 22, 31, 0, 68, 0, 33, 40, 124, 18, 52, 0, 57, 26, 54, 0, 104, 0, 39, 48, 63, 18, 60, 0, 105, 78, 43

Offset: 2

Author

J. M. Bergot and Robert Israel, Dec 04 2020

Keywords

• nonn

Comments

a(n) is the sum of proper prefixes and suffixes of the prime factorization of n.

a(n)=0 if n is prime.

a(n)=p+q if n=p*q is a semiprime.

First differs from A288654 at n=30, with a(30)=28 while A288654(30)=0.

Examples

			12=2*2*3 so a(12) = 2 + 2*2 + 2*3 + 3 = 15.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A288654.

Programs

Maple

f:= proc(n) local L,m;
  L:= sort(map(t -> t[1]$t[2],ifactors(n)[2]));
  m:= nops(L);
  add(mul(L[i],i=1..j)+mul(L[i],i=j+1..m),j=1..m-1)
end proc:
map(f, [$2..100]);

PARI

conv(n) = {my(f=factor(n), v=vector(bigomega(n)), k=1); for (i=1, #f~, for (j=1, f[i,2], v[k] = f[i,1]; k++;);); v;}
a(n) = my(v=conv(n)); sum(j=1, #v-1, prod(k=1, j, v[k])) + sum(j=2, #v, prod(k=j, #v, v[k])); \\ Michel Marcus, Dec 04 2020