A366537 - OEIS

%I #16 Aug 05 2024 16:03:59

%S 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 72,32,48,54,48,50,38,60,56,42,96,44,60,60,72,48,50,78,72,70,54,72,80,

%U 90,60,120,62,96,80,84,144,68,90,96,144,72,74,114,104,100,96

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

%H Amiram Eldar, <a href="/A366537/b366537.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A034448(A004709(n)).	

%F 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 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 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 (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 (Python)

%o from sympy.ntheory.factor_ import udivisor_sigma

%o from sympy import mobius, integer_nthroot

%o def A366537(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 udivisor_sigma(m) # _Chai Wah Wu_, Aug 05 2024

%Y Cf. A002117, A004709, A005117, A034448, A062822, A077610, A358040, A366440, A366536.

%Y Similar sequences: A034676, A366535, A366539.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Oct 12 2023