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

%I A078615 #50 Aug 01 2025 11:07:36
%S A078615 1,4,9,4,25,36,49,4,9,100,121,36,169,196,225,4,289,36,361,100,441,484,
%T A078615 529,36,25,676,9,196,841,900,961,4,1089,1156,1225,36,1369,1444,1521,
%U A078615 100,1681,1764,1849,484,225,2116,2209,36,49,100,2601,676,2809,36,3025,196
%N A078615 a(n) = rad(n)^2, where rad is the squarefree kernel of n (A007947).
%C A078615 It is conjectured that only 1 and 1782 satisfy a(k) = sigma(k). See Broughan link. - _Michel Marcus_, Feb 28 2019
%H A078615 Charles R Greathouse IV, <a href="/A078615/b078615.txt">Table of n, a(n) for n = 1..10000</a>
%H A078615 Peter Bala, <a href="/A368743/a368743.pdf">GCD sum theorems. Two Multivariable Cesaro Type Identities</a>.
%H A078615 K. Broughan, J.-M. De Koninck, I. Kátai, and F. Luca, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Broughan/broughan20.html">On integers for which the sum of divisors is the square of the squarefree core</a>, J. Integer Seq., 15 (2012), pp. 1-12.
%H A078615 Yong-Gao Chen, and Xin Tong, <a href="https://doi.org/10.1016/j.jnt.2015.02.004">On a conjecture of de Koninck</a>, Journal of Number Theory, Volume 154, September 2015, Pages 324-364. Beware of typo 1728.
%F A078615 Multiplicative with a(p^e) = p^2. - _Mitch Harris_, May 17 2005
%F A078615 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
%F A078615 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
%F A078615 a(n) = Sum_{1 <= i, j <= n}  ( mobius(n/gcd(i, j, n)) )^2. - _Peter Bala_, Jan 28 2024
%F A078615 a(n) = Sum_{d|n} mu(d)^2*J_2(d), where J_2 = A007434. - _Ridouane Oudra_, Jul 24 2025
%F A078615 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
%p A078615 a := n -> mul(f,f=map(x->x^2,select(isprime,divisors(n))));
%p A078615 seq(a(n), n=1..56);  # _Peter Luschny_, Mar 30 2014
%t A078615 a[n_] := Times @@ FactorInteger[n][[All, 1]]^2; Array[a, 60] (* _Jean-François Alcover_, Jun 04 2019 *)
%o A078615 (PARI) a(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])^2 \\ _Charles R Greathouse IV_, Aug 06 2013
%Y A078615 Cf. A002117, A007948, A055231, A062378, A330523, A007434, A008683, A001221, A065958.
%K A078615 nonn,mult,easy
%O A078615 1,2
%A A078615 _Reinhard Zumkeller_, Dec 10 2002