A063958 Sum of the non-unitary prime divisors of n: sum of those prime divisors for which the exponent in the prime factorization exceeds 1.
0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 2, 0, 0, 0, 2, 0, 3, 0, 2, 0, 0, 0, 2, 5, 0, 3, 2, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 2, 7, 5, 0, 2, 0, 3, 0, 2, 0, 0, 0, 2, 0, 0, 3, 2, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 5, 2, 0, 0, 0, 2, 3, 0, 0, 2, 0, 0, 0, 2, 0, 3, 0, 2, 0, 0, 0, 2, 0, 7, 3, 7, 0, 0, 0, 2, 0
Offset: 1
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from Harry J. Smith)
Programs
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Maple
a:= proc(n) option remember; add(`if`(i[2]>1, i[1], 0), i=ifactors(n)[2]) end: seq(a(n), n=1..100); # Alois P. Heinz, Jun 24 2018
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Mathematica
Array[Total@ Select[FactorInteger@ #, Last@ # > 1 &][[All, 1]] &, 105] (* Michael De Vlieger, Dec 06 2018 *)
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PARI
{ for (n=1, 1000, f=factor(n)~; a=0; for (i=1, length(f), if (f[2, i]>1, a+=f[1, i])); write("b063958.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 04 2009
Formula
G.f.: Sum_{k>=1} prime(k) * x^(prime(k)^2) / (1 - x^(prime(k)^2)). - Ilya Gutkovskiy, Apr 06 2020
a(n) = sopf(rad(n/rad(n))). - Wesley Ivan Hurt, Nov 21 2021
a(n) = Sum_{p^2|n} p. - Wesley Ivan Hurt, Feb 21 2022
From Amiram Eldar, Jul 24 2024: (Start)
Additive with a(p^e) = p if e >= 2, and 0 otherwise. (End)