A300906 Numbers k such that sigma(k)^k divides k^sigma(k).

1, 6, 28, 84, 120, 364, 420, 496, 672, 840, 1080, 1320, 1488, 1782, 2280, 2760, 3276, 3360, 3472, 3480, 3720, 3780, 5640, 7080, 7392, 7440, 7560, 8128, 8736, 9240, 9480, 10416, 10920, 11880, 12400, 15456, 15960, 16368, 16380, 17880, 18360, 18600, 19320, 20520

Offset: 1

Author

Jaroslav Krizek, Mar 20 2018

Keywords

• nonn

Comments

Numbers k such that A217872(k) divides A100879(k).

Numbers k such that A300905(k) = 0.

Corresponding quotients: 1, 729, 123476695691247935826229781856256, ...

m-perfect numbers k (A007691) are terms iff m divides k.

Examples

			6 is a term because 6^sigma(6) / sigma(6)^6 = 6^12 / 12^6 = 2176782336 / 2985984 = 729 (integer).
		

Crossrefs

Cf. A000203, A100879, A217872, A300905.

Programs

GAP

Filtered([1..30000],n->PowerModInt(n,Sigma(n),Sigma(n)^n)=0); # Muniru A Asiru, Mar 20 2018

Magma

[n: n in[1..20000]  | n^SumOfDivisors(n) mod SumOfDivisors(n)^n eq 0];

Maple

with(numtheory):
select(n->n &^ sigma(n) mod sigma(n)^n =0, [`$`(1..30000)]); # Muniru A Asiru, Mar 20 2018

PARI

isok(n) = my(s = sigma(n)); Mod(n, s^n)^s == 0; \\ Michel Marcus, Mar 23 2018