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A366440 The sum of divisors of the cubefree numbers (A004709).

%I A366440 #12 Aug 06 2024 02:11:06
%S A366440 1,3,4,7,6,12,8,13,18,12,28,14,24,24,18,39,20,42,32,36,24,31,42,56,30,
%T A366440 72,32,48,54,48,91,38,60,56,42,96,44,84,78,72,48,57,93,72,98,54,72,80,
%U A366440 90,60,168,62,96,104,84,144,68,126,96,144,72,74,114,124,140
%N A366440 The sum of divisors of the cubefree numbers (A004709).
%H A366440 Amiram Eldar, <a href="/A366440/b366440.txt">Table of n, a(n) for n = 1..10000</a>
%H A366440 Rafael Jakimczuk and Matilde Lalín, <a href="https://doi.org/10.7546/nntdm.2022.28.4.617-634">Asymptotics of sums of divisor functions over sequences with restricted factorization structure</a>, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634; p. 619, eq. (2).
%F A366440 a(n) = A000203(A004709(n)).
%F A366440 Sum_{k=1..n} a(k) ~ c * n^2, where c = 15*zeta(3)/(2*Pi^2) = A082020 * A002117 / 2 = 0.913453711751... .
%F A366440 The asymptotic mean of the abundancy index of the cubefree numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A004709(k) = 15/Pi^2 = 1.519817... (A082020).
%t A366440 f[p_, e_] := (p^(e+1)-1)/(p-1); s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;;, 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; s[1] = 1; Array[s, 100]
%o A366440 (PARI) lista(max) = for(k = 1, max, my(f = factor(k), iscubefree = 1); for(i = 1, #f~, if(f[i, 2] > 2, iscubefree = 0; break)); if(iscubefree, print1(sigma(f), ", ")));
%o A366440 (Python)
%o A366440 from sympy import mobius, integer_nthroot, divisor_sigma
%o A366440 def A366440(n):
%o A366440     def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
%o A366440     m, k = n, f(n)
%o A366440     while m != k:
%o A366440         m, k = k, f(k)
%o A366440     return divisor_sigma(m) # _Chai Wah Wu_, Aug 06 2024
%Y A366440 Cf. A000203, A004709, A358040.
%Y A366440 Cf. A002117, A082020.
%Y A366440 Similar sequences: A062822, A065764, A180114, A362986, A366439.
%K A366440 nonn,easy
%O A366440 1,2
%A A366440 _Amiram Eldar_, Oct 10 2023