%I #16 Sep 08 2022 08:46:23

%S 1,6,12,28,56,120,360,496,672,992,2016,8128,16256,30240,32760,60480,

%T 65520,120960,131040,523776,1571328,2178540,4357080,8714160,23569920,

%U 33550336,45532800,47139840,67100672,91065600,94279680,142990848,182131200,285981696

%N Numbers m having at least one divisor d such that m divides sigma(d).

%C Generalization of multiperfect numbers (A007691).

%C Multiperfect numbers (A007691) are terms. If m is a k-multiperfect number and d divides k (for k > 1 and d > 1), then d*m is also a term.

%C Number 1379454720 is the smallest number with two divisors d with this property (459818240 and 1379454720). Another such number is 153003540480 with divisors 51001180160 and 153003540480. Is there a number with three divisors d with this property?

%C Supersequence of A081756.

%e 12 is a term because 6 divides 12 and simultaneously 12 divides sigma(6) = 12.

%t Select[Range[530000],AnyTrue[DivisorSigma[1,Divisors[#]]/#,IntegerQ]&] (* The program generates the first 20 terms of the sequence. To generate more, increase the Range constant, but the program may take a long time to run. *) (* _Harvey P. Dale_, Jan 17 2022 *)

%o (Magma) [n: n in [1..10000] | #[d: d in Divisors(n) | SumOfDivisors(d) mod n eq 0] gt 0]

%o (PARI) isok(n) = {fordiv(n, d, if (!(sigma(d) % n), return (1));); return (0);} \\ _Michel Marcus_, Jan 21 2019

%Y Cf. A000203, A007691, A081756, A323653.

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Jan 21 2019