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A073245 Sum of all cubefree numbers with the same squarefree kernel as the n-th squarefree number. 2

%I #24 Oct 09 2023 02:20:00

%S 1,6,12,30,72,56,180,132,182,336,360,306,380,672,792,552,1092,870,

%T 2160,992,1584,1836,1680,1406,2280,2184,1722,4032,1892,3312,2256,3672,

%U 2862,3960,4560,5220,3540,3782,5952,5460,9504,4556,6624,10080,5112,5402

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

%H Amiram Eldar, <a href="/A073245/b073245.txt">Table of n, a(n) for n = 1..10000</a>

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

%F Sum_{n>=1} 1/a(n) = A306633. - _Amiram Eldar_, Oct 14 2020

%F a(n) = A064987(A005117(n)). - _Michel Marcus_, Oct 18 2020

%F 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 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 Map[# * DivisorSigma[1, #] &, Select[Range[200], SquareFreeQ]] (* _Amiram Eldar_, Oct 14 2020 *)

%o (PARI) apply(x->(x*sigma(x)), select(issquarefree, [1..100])) \\ _Michel Marcus_, Oct 18 2020

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

%Y Cf. A002117, A013661.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Jul 21 2002

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)