cp's OEIS Frontend

A385131 The sum of divisors of n whose maximum exponent in their prime factorization is odd.

%I A385131 #10 Jun 24 2025 06:15:22
%S A385131 0,2,3,2,5,11,7,10,3,17,11,11,13,23,23,10,17,11,19,17,31,35,23,43,5,
%T A385131 41,30,23,29,71,31,42,47,53,47,11,37,59,55,65,41,95,43,35,23,71,47,43,
%U A385131 7,17,71,41,53,92,71,87,79,89,59,71,61,95,31,42,83,143,67,53
%N A385131 The sum of divisors of n whose maximum exponent in their prime factorization is odd.
%C A385131 The sum of divisors of n that are not terms in A368714.
%C A385131 The number of these divisors is A385129(n).
%H A385131 Amiram Eldar, <a href="/A385131/b385131.txt">Table of n, a(n) for n = 1..10000</a>
%F A385131 a(n) = Sum_{d|n} (d * (A051903(d) mod 2)).
%F A385131 a(n) = A000203(n) - A385130(n).
%F A385131 a(n) = Sum_{k=1..kmax(n)} (-1)^k * Product_{i=1..r} (p_i^(min(e_i, k-1) + 1)-1)/(p_i-1), for n >= 2; if n = Product_{i=1..r} p_i^e_i, r = omega(n) = A001221(n), then emax(n) = max(e_i) = A051903(n), and kmax(n) = emax(n)+1 if emax(n) is odd, and emax(n) otherwise.
%F A385131 Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = 1 - Sum_{k>=2} (-1)^k * (1-1/zeta(k)) = 0.7240832794017729923099...,
%t A385131 q[n_] := OddQ[Max[FactorInteger[n][[;; , 2]]]]; q[1] = False; a[n_] := DivisorSum[n, # &, q[#] &]; Array[a, 100]		
%t A385131 (* second program: *)
%t A385131 a[n_] := Module[{f = FactorInteger[n], p, e, emax, kmax}, p = f[[;;,1]]; e = f[[;;,2]]; emax = Max[e]; kmax = emax + Mod[emax, 2]; Sum[(-1)^k * Product[(p[[i]]^(Min[e[[i]], k-1]+1)-1)/(p[[i]]-1), {i, 1, Length[e]}], {k, 1, kmax}]]; a[1] = 0; Array[a, 100]
%o A385131 (PARI) q(n) = if(n == 1, 0, vecmax(factor(n)[,2]) % 2);
%o A385131 a(n) = sumdiv(n, d, d*q(d));
%o A385131 (PARI) a(n) = if(n == 1, 0, my(f = factor(n), p = f[,1], e = f[,2], emax = vecmax(e), kmax = emax + emax % 2); sum(k = 1, kmax, (-1)^k * prod(i = 1, #e, (p[i]^(min(e[i], k-1)+1)-1)/(p[i]-1))));
%Y A385131 Cf. A000203, A001221, A013661, A051903, A368714, A383156, A385129, A385130.
%K A385131 nonn,easy
%O A385131 1,2
%A A385131 _Amiram Eldar_, Jun 24 2025