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A076290
Sum of the semiprime divisors of n.
15
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
Sequence in context: A210625 A210615 A179312 * A198224 A178105 A178109
KEYWORD
easy,nonn
AUTHOR
Joseph L. Pe, Nov 24 2002
STATUS
approved