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

1, 6, 12, 30, 72, 56, 180, 132, 182, 336, 360, 306, 380, 672, 792, 552, 1092, 870, 2160, 992, 1584, 1836, 1680, 1406, 2280, 2184, 1722, 4032, 1892, 3312, 2256, 3672, 2862, 3960, 4560, 5220, 3540, 3782, 5952, 5460, 9504, 4556, 6624, 10080, 5112, 5402

Offset: 1

Author

Reinhard Zumkeller, Jul 21 2002

Keywords

• nonn

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A004709, A005117, A007947, A062822, A064987, A072048, A306633.

Cf. A002117, A013661.

Programs

Mathematica

Map[# * DivisorSigma[1, #] &, Select[Range[200], SquareFreeQ]] (* Amiram Eldar, Oct 14 2020 *)

PARI

apply(x->(x*sigma(x)), select(issquarefree, [1..100])) \\ Michel Marcus, Oct 18 2020

Python

from math import isqrt
from sympy import mobius, divisor_sigma
def A073245(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m*divisor_sigma(m) # Chai Wah Wu, Aug 12 2024

Formula

a(n) = A062822(n)*A005117(n).

Sum_{n>=1} 1/a(n) = A306633. - Amiram Eldar, Oct 14 2020

a(n) = A064987(A005117(n)). - Michel Marcus, Oct 18 2020

Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)^3/(3*zeta(3)) = 1.23423882415851340020... . - Amiram Eldar, Oct 09 2023