A076290 Sum of the semiprime divisors of n.

0, 0, 0, 4, 0, 6, 0, 4, 9, 10, 0, 10, 0, 14, 15, 4, 0, 15, 0, 14, 21, 22, 0, 10, 25, 26, 9, 18, 0, 31, 0, 4, 33, 34, 35, 19, 0, 38, 39, 14, 0, 41, 0, 26, 24, 46, 0, 10, 49, 35, 51, 30, 0, 15, 55, 18, 57, 58, 0, 35, 0, 62, 30, 4, 65, 61, 0, 38, 69, 59, 0, 19, 0, 74, 40, 42, 77, 71, 0

Offset

1,4

Comments

A semiprime is a product of two primes.

Links

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|n} d * [Omega(d) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021

Example

The divisors of 12 are 1, 2, 3, 4, 6, 12, of which 4 and 6 are semiprime. Hence a(12) = 4 + 6 = 10.

Maple

a:= proc(n) local l, m; l:=ifactors(n)[2]; m:=nops(l);
      add(`if`(l[i][2]>1, l[i][1]^2, 0)+
      add(l[i][1]*l[j][1], j=i+1..m), i=1..m)
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Jul 18 2013

Mathematica

isSP[n_] := Module[{f, l}, f = FactorInteger[n]; l = Length[f]; (l == 2 && f[[1]][[2]] == 1 && f[[2]][[2]] == 1) || (l == 1 && f[[1]][[2]] == 2)]; f[n_] := Module[{a, d, l}, a = {}; d = Divisors[n]; l = Length[d]; For[i = 1, i <= l, i++, If[isSP[d[[i]]], a = Append[a, d[[i]]]]]; a]; Table[Apply[Plus, f[i]], {i, 1, 100}]

Prog

(PARI) a(n) = local(fn, r, om); fn=factor(n); r=om=0; for(i=1, matsize(fn)[1], om+=fn[i, 1]; r+=fn[i, 1]^2*if(fn[i, 2]==1, -1, 1)); (r+om^2)\2 \\ Franklin T. Adams-Watters, Jul 26 2009

Crossrefs

Cf. A001222, A001358.

Sequence in context: A210625 A210615 A179312 * A198224 A178105 A178109

Adjacent sequences: A076287 A076288 A076289 * A076291 A076292 A076293

Keyword

easy,nonn

Author

Joseph L. Pe, Nov 24 2002

Status

approved