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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.

%I A063958 #34 Jul 25 2024 03:22:12
%S A063958 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,
%T A063958 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,
%U A063958 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
%N 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.
%H A063958 Alois P. Heinz, <a href="/A063958/b063958.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from Harry J. Smith)
%F A063958 G.f.: Sum_{k>=1} prime(k) * x^(prime(k)^2) / (1 - x^(prime(k)^2)). - _Ilya Gutkovskiy_, Apr 06 2020
%F A063958 a(n) = sopf(rad(n/rad(n))). - _Wesley Ivan Hurt_, Nov 21 2021
%F A063958 a(n) = Sum_{p^2|n} p. - _Wesley Ivan Hurt_, Feb 21 2022
%F A063958 From _Amiram Eldar_, Jul 24 2024: (Start)
%F A063958 a(n) = A008472(n) - A063956(n).
%F A063958 Additive with a(p^e) = p if e >= 2, and 0 otherwise. (End)
%p A063958 a:= proc(n) option remember; add(`if`(i[2]>1, i[1], 0), i=ifactors(n)[2]) end:
%p A063958 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jun 24 2018
%t A063958 Array[Total@ Select[FactorInteger@ #, Last@ # > 1 &][[All, 1]] &, 105] (* _Michael De Vlieger_, Dec 06 2018 *)
%o A063958 (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
%Y A063958 Cf. A034444, A048105, A056169, A056170, A063956.
%Y A063958 Cf. A007947 (rad), A008472 (sopf).
%K A063958 nonn,easy
%O A063958 1,4
%A A063958 _Labos Elemer_, Sep 04 2001