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Sum of the semiprime divisors of n.
15

%I #13 Jan 29 2021 06:40:17

%S 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,

%T 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,

%U 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

%N Sum of the semiprime divisors of n.

%C A semiprime is a product of two primes.

%H Franklin T. Adams-Watters, <a href="/A076290/b076290.txt">Table of n, a(n) for n = 1..10000</a>

%F 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

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

%p a:= proc(n) local l, m; l:=ifactors(n)[2]; m:=nops(l);

%p add(`if`(l[i][2]>1, l[i][1]^2, 0)+

%p add(l[i][1]*l[j][1], j=i+1..m), i=1..m)

%p end:

%p seq(a(n), n=1..120); # _Alois P. Heinz_, Jul 18 2013

%t 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}]

%o (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

%Y Cf. A001222, A001358.

%K easy,nonn

%O 1,4

%A _Joseph L. Pe_, Nov 24 2002