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A366537 The sum of unitary divisors of the cubefree numbers (A004709).

%I A366537 #19 Apr 26 2025 21:35:05
%S A366537 1,3,4,5,6,12,8,10,18,12,20,14,24,24,18,30,20,30,32,36,24,26,42,40,30,
%T A366537 72,32,48,54,48,50,38,60,56,42,96,44,60,60,72,48,50,78,72,70,54,72,80,
%U A366537 90,60,120,62,96,80,84,144,68,90,96,144,72,74,114,104,100,96
%N A366537 The sum of unitary divisors of the cubefree numbers (A004709).
%H A366537 Amiram Eldar, <a href="/A366537/b366537.txt">Table of n, a(n) for n = 1..10000</a>
%F A366537 a(n) = A034448(A004709(n)).
%F A366537 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(3)^2 * Product_{p prime} (1 + 1/p^2 - 2/p^3 + 1/p^4 - 1/p^5) = 1.665430860774244601005... .
%F A366537 The asymptotic mean of the unitary abundancy index of the cubefree numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A004709(k) = c / zeta(3) = 1.38548421160152785073... .
%t A366537 s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, # < 3 &], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]
%o A366537 (PARI) lista(max) = for(k = 1, max, my(f = factor(k), e = f[, 2], iscubefree = 1); for(i = 1, #e, if(e[i] > 2, iscubefree = 0; break)); if(iscubefree, print1(prod(i = 1, #e, 1 + f[i, 1]^e[i]), ", ")));
%o A366537 (Python)
%o A366537 from sympy.ntheory.factor_ import udivisor_sigma
%o A366537 from sympy import mobius, integer_nthroot
%o A366537 def A366537(n):
%o A366537     def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
%o A366537     m, k = n, f(n)
%o A366537     while m != k:
%o A366537         m, k = k, f(k)
%o A366537     return udivisor_sigma(m) # _Chai Wah Wu_, Aug 05 2024
%Y A366537 Cf. A002117, A004709, A005117, A034448, A062822, A077610, A358040, A366440, A366536.
%Y A366537 Similar sequences: A034676, A366535, A366539.
%K A366537 nonn,easy
%O A366537 1,2
%A A366537 _Amiram Eldar_, Oct 12 2023