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).

0, 1, 5, -3, 19, 24, 41, -11, -4, 82, 109, 8, 155, 172, 201, -27, 271, -3, 341, 58, 409, 448, 505, -24, -6, 634, -31, 140, 811, 828, 929, -59, 1041, 1102, 1177, -55, 1331, 1384, 1465, 10, 1639, 1668, 1805, 400, 147, 2044, 2161, -88, -8, 7, 2529, 578, 2755, -84, 2953

Offset: 1

Author

Michel Marcus, Nov 09 2020

Keywords

• sign

Comments

It is conjectured that only 1 and 1782 satisfy a(x) = 0.

References

• R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B11.

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000
• K. Broughan, J.-M. De Koninck, I. Kátai, 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.

Crossrefs

Cf. A000203, A007947, A078615, A326142.

Programs

Maple

a:= n-> mul(i[1], i=ifactors(n)[2])^2-numtheory[sigma](n):
seq(a(n), n=1..60);  # Alois P. Heinz, Nov 09 2020

PARI

a(n) = my(f=factor(n)); factorback(f[, 1])^2 - sigma(f);

Formula

a(n) = A007947(n)^2 - A000203(n).

a(n) = A078615(n) - A000203(n).