cp's OEIS Frontend

A073245 Sum of all cubefree numbers with the same squarefree kernel as the n-th squarefree number.

%I A073245 #29 Aug 13 2024 02:21:08
%S A073245 1,6,12,30,72,56,180,132,182,336,360,306,380,672,792,552,1092,870,
%T A073245 2160,992,1584,1836,1680,1406,2280,2184,1722,4032,1892,3312,2256,3672,
%U A073245 2862,3960,4560,5220,3540,3782,5952,5460,9504,4556,6624,10080,5112,5402
%N A073245 Sum of all cubefree numbers with the same squarefree kernel as the n-th squarefree number.
%H A073245 Amiram Eldar, <a href="/A073245/b073245.txt">Table of n, a(n) for n = 1..10000</a>
%F A073245 a(n) = A062822(n)*A005117(n).
%F A073245 Sum_{n>=1} 1/a(n) = A306633. - _Amiram Eldar_, Oct 14 2020
%F A073245 a(n) = A064987(A005117(n)). - _Michel Marcus_, Oct 18 2020
%F A073245 Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)^3/(3*zeta(3)) = 1.23423882415851340020... . - _Amiram Eldar_, Oct 09 2023
%e A073245 14 is the 10th squarefree number: A005117(10)=14=2*7, the cubefree numbers with squarefree kernel =14 are 14, 28=2*2*7, 98=2*7*7 and 196=2*2*7*7; therefore a(10)=14+28+98+196=336; a(10)=A062822(10)*A005117(10)=24*14=336.
%t A073245 Map[# * DivisorSigma[1, #] &, Select[Range[200], SquareFreeQ]] (* _Amiram Eldar_, Oct 14 2020 *)
%o A073245 (PARI) apply(x->(x*sigma(x)), select(issquarefree, [1..100])) \\ _Michel Marcus_, Oct 18 2020
%o A073245 (Python)
%o A073245 from math import isqrt
%o A073245 from sympy import mobius, divisor_sigma
%o A073245 def A073245(n):
%o A073245     def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A073245     m, k = n, f(n)
%o A073245     while m != k:
%o A073245         m, k = k, f(k)
%o A073245     return m*divisor_sigma(m) # _Chai Wah Wu_, Aug 12 2024
%Y A073245 Cf. A004709, A005117, A007947, A062822, A064987, A072048, A306633.
%Y A073245 Cf. A002117, A013661.
%K A073245 nonn
%O A073245 1,2
%A A073245 _Reinhard Zumkeller_, Jul 21 2002