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

1,4

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from Harry J. Smith)

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)

a(n) = A008472(n) - A063956(n).

Additive with a(p^e) = p if e >= 2, and 0 otherwise. (End)

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

Mathematica

Array[Total@ Select[FactorInteger@ #, Last@ # > 1 &][[All, 1]] &, 105] (* Michael De Vlieger, Dec 06 2018 *)

Prog

(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

Crossrefs

Cf. A034444, A048105, A056169, A056170, A063956.

Cf. A007947 (rad), A008472 (sopf).

Sequence in context: A263145 A057108 A349435 * A349434 A126164 A340317

Adjacent sequences: A063955 A063956 A063957 * A063959 A063960 A063961

Keyword

nonn,easy

Author

Labos Elemer, Sep 04 2001

Status

approved