A078615 a(n) = rad(n)^2, where rad is the squarefree kernel of n (A007947).

1, 4, 9, 4, 25, 36, 49, 4, 9, 100, 121, 36, 169, 196, 225, 4, 289, 36, 361, 100, 441, 484, 529, 36, 25, 676, 9, 196, 841, 900, 961, 4, 1089, 1156, 1225, 36, 1369, 1444, 1521, 100, 1681, 1764, 1849, 484, 225, 2116, 2209, 36, 49, 100, 2601, 676, 2809, 36, 3025, 196

Offset

1,2

Comments

It is conjectured that only 1 and 1782 satisfy a(k) = sigma(k). See Broughan link. - Michel Marcus, Feb 28 2019

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

K. Broughan, J.-M. De Koninck, I. Kátai, and F. Luca, On integers for which the sum of divisors is the square of the squarefree core, J. Integer Seq., 15 (2012), pp. 1-12.

Yong-Gao Chen, and Xin Tong, On a conjecture of de Koninck, Journal of Number Theory, Volume 154, September 2015, Pages 324-364. Beware of typo 1728.

Formula

Multiplicative with a(p^e) = p^2. - Mitch Harris, May 17 2005

G.f.: Sum_{k>=1} mu(k)^2*J_2(k)*x^k/(1 - x^k), where J_2() is the Jordan function. - Ilya Gutkovskiy, Nov 06 2018

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A002117 * A330523 / 3 = 0.214725... . - Amiram Eldar, Oct 30 2022

a(n) = Sum_{1 <= i, j <= n} ( mobius(n/gcd(i, j, n)) )^2. - Peter Bala, Jan 28 2024

Maple

a := n -> mul(f, f=map(x->x^2, select(isprime, divisors(n))));
seq(a(n), n=1..56);  # Peter Luschny, Mar 30 2014

Mathematica

a[n_] := Times @@ FactorInteger[n][[All, 1]]^2; Array[a, 60] (* Jean-François Alcover, Jun 04 2019 *)

Prog

(PARI) a(n)=my(f=factor(n)[, 1]); prod(i=1, #f, f[i])^2 \\ Charles R Greathouse IV, Aug 06 2013

Crossrefs

Cf. A002117, A007948, A055231, A062378, A330523.

Sequence in context: A005063 A235323 A345304 * A197863 A087320 A087321

Adjacent sequences: A078612 A078613 A078614 * A078616 A078617 A078618

Keyword

nonn,mult,easy

Author

Reinhard Zumkeller, Dec 10 2002

Status

approved