cp's OEIS Frontend

A286875 If n = Product (p_j^k_j) then a(n) = Sum (k_j >= 2, p_j^k_j).

%I A286875 #25 Jul 25 2024 03:12:27
%S A286875 0,0,0,4,0,0,0,8,9,0,0,4,0,0,0,16,0,9,0,4,0,0,0,8,25,0,27,4,0,0,0,32,
%T A286875 0,0,0,13,0,0,0,8,0,0,0,4,9,0,0,16,49,25,0,4,0,27,0,8,0,0,0,4,0,0,9,
%U A286875 64,0,0,0,4,0,0,0,17,0,0,25,4,0,0,0,16,81,0,0,4,0,0,0,8,0,9,0,4,0,0,0,32,0,49,9,29,0,0,0,8,0,0,0,31
%N A286875 If n = Product (p_j^k_j) then a(n) = Sum (k_j >= 2, p_j^k_j).
%C A286875 Sum of unitary, proper prime power divisors of n.
%H A286875 Antti Karttunen, <a href="/A286875/b286875.txt">Table of n, a(n) for n = 1..16384</a>
%H A286875 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F A286875 a(n) = Sum_{d|n, d = p^k, p prime, k >= 2, gcd(d, n/d) = 1} d.
%F A286875 a(A246547(k)) = A246547(k).
%F A286875 a(A005117(k)) = 0.
%F A286875 Additive with a(p^e) = p^e if e >= 2, and 0 otherwise. - _Amiram Eldar_, Jul 24 2024
%e A286875 a(360) = a(2^3*3^2*5) = 2^3 + 3^2 = 17.
%t A286875 Table[DivisorSum[n, # &, CoprimeQ[#, n/#] && PrimePowerQ[#] && !PrimeQ[#] &], {n, 108}]
%t A286875 f[p_, e_] := If[e == 1, 0, p^e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jul 24 2024 *)
%o A286875 (Python)
%o A286875 from sympy import primefactors, isprime, gcd, divisors
%o A286875 def a(n): return sum(d for d in divisors(n) if gcd(d, n//d)==1 and len(primefactors(d))==1 and not isprime(d))
%o A286875 print([a(n) for n in range(1, 109)]) # _Indranil Ghosh_, Aug 02 2017
%o A286875 (PARI) A286875(n) = { my(f=factor(n)); for (i=1, #f~, if(f[i, 2] < 2, f[i, 1] = 0)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); }; \\ _Antti Karttunen_, Oct 07 2017
%Y A286875 Cf. A005117, A008475, A222416, A023888, A023889, A034448, A063956, A077610, A092261, A246547, A284117.
%K A286875 nonn,easy
%O A286875 1,4
%A A286875 _Ilya Gutkovskiy_, Aug 02 2017