A046987 Multiply perfect numbers that are neither harmonic numbers nor arithmetic numbers.

120, 523776, 1476304896, 31998395520, 30823866178560, 69357059049509038080, 4010059765937523916800, 27099073228001299660800, 686498980761986918441287680, 2827987212986831882236723200, 115131961034430181728489308160, 13361233986454282110797768294400, 32789312424503984621373515366400

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Examples

			k = 523776 is a term since s0 = d(k) = 80, s1 = sigma(k) = 1571328, s1/k = 1571328/523776 = 3 is an integer, but (k * s0)/s1 = 80/3 and s1/s0 = 98208/5 are not integers.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..100

Crossrefs

In A007691 but neither in A003601 nor in A001599.

Cf. A000005, A000203, A046985, A046986.

Programs

Mathematica

q[n_] := Module[{d = DivisorSigma[0, n], s = DivisorSigma[1, n]}, Divisible[s, n] && !Divisible[n * d, s] && !Divisible[s, d]]; Select[Range[6*10^5], q] (* Amiram Eldar, May 09 2024 *)

PARI

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && ((k * d) % s) && (s % d);} \\ Amiram Eldar, May 09 2024

Formula

Let s1 be the sum of divisors of k and s0 be the number of divisors of k. Then, k is a term if k | s1, but (k * s0) is not divisible by s1, and s1 is not divisible by s0.

Extensions

a(5)-a(10) from Donovan Johnson, Nov 30 2008

Edited and a(11)-a(13) added by Amiram Eldar, May 09 2024