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

1,1

Links

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

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.

Example

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.

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 *)

Prog

(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

Crossrefs

In A007691 but neither in A003601 nor in A001599.

Cf. A000005, A000203, A046985, A046986.

Sequence in context: A003789 A325024 A325026 * A127232 A208191 A166351

Adjacent sequences: A046984 A046985 A046986 * A046988 A046989 A046990

Keyword

nonn

Author

Labos Elemer

Extensions

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

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

Status

approved