A366440 - OEIS

%I #12 Aug 06 2024 02:11:06

%S 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 72,32,48,54,48,91,38,60,56,42,96,44,84,78,72,48,57,93,72,98,54,72,80,

%U 90,60,168,62,96,104,84,144,68,126,96,144,72,74,114,124,140

%N The sum of divisors of the cubefree numbers (A004709).

%H Amiram Eldar, <a href="/A366440/b366440.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 a(n) = A000203(A004709(n)).

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 15*zeta(3)/(2*Pi^2) = A082020 * A002117 / 2 = 0.913453711751... .

%F 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 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 (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 (Python)

%o from sympy import mobius, integer_nthroot, divisor_sigma

%o def A366440(n):

%o     def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))

%o     m, k = n, f(n)

%o     while m != k:

%o         m, k = k, f(k)

%o     return divisor_sigma(m) # _Chai Wah Wu_, Aug 06 2024

%Y Cf. A000203, A004709, A358040.

%Y Cf. A002117, A082020.

%Y Similar sequences: A062822, A065764, A180114, A362986, A366439.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Oct 10 2023