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

1,2

Links

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

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

Cf. A002117, A013661.

Sequence in context: A236539 A122211 A015801 * A119626 A152522 A096356

Adjacent sequences: A073242 A073243 A073244 * A073246 A073247 A073248

Keyword

nonn

Author

Reinhard Zumkeller, Jul 21 2002

Status

approved