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Numbers k such that sigma(k)^k divides k^sigma(k).
3

%I #17 Sep 08 2022 08:46:20

%S 1,6,28,84,120,364,420,496,672,840,1080,1320,1488,1782,2280,2760,3276,

%T 3360,3472,3480,3720,3780,5640,7080,7392,7440,7560,8128,8736,9240,

%U 9480,10416,10920,11880,12400,15456,15960,16368,16380,17880,18360,18600,19320,20520

%N Numbers k such that sigma(k)^k divides k^sigma(k).

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

%C Numbers k such that A300905(k) = 0.

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

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

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

%p with(numtheory):

%p select(n->n &^ sigma(n) mod sigma(n)^n =0, [`$`(1..30000)]); # _Muniru A Asiru_, Mar 20 2018

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

%o (GAP) Filtered([1..30000],n->PowerModInt(n,Sigma(n),Sigma(n)^n)=0); # _Muniru A Asiru_, Mar 20 2018

%o (PARI) isok(n) = my(s = sigma(n)); Mod(n, s^n)^s == 0; \\ _Michel Marcus_, Mar 23 2018

%Y Cf. A000203, A100879, A217872, A300905.

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Mar 20 2018