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A338790 a(n) = rad(n)^2 - sigma(n), where rad(n) is the squarefree kernel of n (A007947) and sigma(n) is the sum of divisors of n (A000203).

%I A338790 #12 Nov 09 2020 14:21:37
%S A338790 0,1,5,-3,19,24,41,-11,-4,82,109,8,155,172,201,-27,271,-3,341,58,409,
%T A338790 448,505,-24,-6,634,-31,140,811,828,929,-59,1041,1102,1177,-55,1331,
%U A338790 1384,1465,10,1639,1668,1805,400,147,2044,2161,-88,-8,7,2529,578,2755,-84,2953
%N A338790 a(n) = rad(n)^2 - sigma(n), where rad(n) is the squarefree kernel of n (A007947) and sigma(n) is the sum of divisors of n (A000203).
%C A338790 It is conjectured that only 1 and 1782 satisfy a(x) = 0.
%D A338790 R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B11.
%H A338790 Michel Marcus, <a href="/A338790/b338790.txt">Table of n, a(n) for n = 1..10000</a>
%H A338790 K. Broughan, J.-M. De Koninck, I. Kátai, 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 A338790 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 A338790 a(n) =  A007947(n)^2 - A000203(n).
%F A338790 a(n) =  A078615(n) - A000203(n).
%p A338790 a:= n-> mul(i[1], i=ifactors(n)[2])^2-numtheory[sigma](n):
%p A338790 seq(a(n), n=1..60);  # _Alois P. Heinz_, Nov 09 2020
%o A338790 (PARI) a(n) = my(f=factor(n)); factorback(f[, 1])^2 - sigma(f);
%Y A338790 Cf. A000203, A007947, A078615, A326142.
%K A338790 sign
%O A338790 1,3
%A A338790 _Michel Marcus_, Nov 09 2020