A263928 Integers m such that sigma(m)^2 is divisible by m.

1, 6, 24, 28, 120, 224, 234, 270, 496, 588, 600, 672, 864, 1080, 1521, 1638, 1782, 2016, 3724, 4320, 4680, 5733, 6048, 6200, 6552, 7128, 8128, 11172, 11466, 15872, 17280, 18144, 18600, 18620, 21600, 22932, 26208, 26460, 27000, 30240, 32640, 32760, 33516, 35640

Offset: 1

Author

Paolo P. Lava, Oct 30 2015

Keywords

• nonn

Comments

Previous name was: "Numbers such that the product of the sum of their divisors and the sum of the reciprocals of their divisors is an integer".

The multiply-perfect numbers (A007691) are a subset of this sequence.

This is a subsequence of A175200. - Michel Marcus, Nov 03 2015

Alternative definition: Numbers m such that Sum_{i = 1..k} (sigma(m) - d_i) / d_i is an integer, where d_i are the k divisors of m. - Paolo P. Lava, Mar 23 2017

Examples

			Divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Their sum is sigma(24) = 60 while the sum of their reciprocals is 1/1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/12 + 1/24 = 5/2. Finally 60 * 5/2 = 150.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A000203, A007691, A017665, A017666, A175200, A263983.

Programs

Maple

with(numtheory): P:=proc(q) local n;
for n from 1 to q do if type(sigma(n)^2/n,integer) then print(n);
fi; od; end: P(10^6);

Mathematica

Select[Range[36000],Divisible[DivisorSigma[1,#]^2,#]&] (* Harvey P. Dale, Jul 05 2023 *)

PARI

isok(n) = (sigma(n)^2 % n) == 0; \\ Michel Marcus, Nov 03 2015

Extensions

New name from Michel Marcus, Nov 03 2015