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
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.
Programs
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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
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Mathematica
a[n_] := Times @@ FactorInteger[n][[All, 1]]^2; Array[a, 60] (* Jean-François Alcover, Jun 04 2019 *)
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PARI
a(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])^2 \\ Charles R Greathouse IV, Aug 06 2013
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
a(n) = Sum_{d|n} mu(d)^2*J_2(d), where J_2 = A007434. - Ridouane Oudra, Jul 24 2025
a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*Psi_2(d), where omega = A001221 and Psi_2 = A065958. - Ridouane Oudra, Aug 01 2025
Comments