A056075 Numbers m such that m divides sigma(m) - d(m).

1, 4, 56, 7192, 7232, 7912, 10792, 17272, 30592, 114256, 2154584, 3428368, 44375136, 89245784, 2739393699744, 36993585958528, 47319950478240, 118122891971648, 943226995376128, 2737657760695168, 5020331545072768, 36028789368553472, 40256362055287184, 42381542060395136, 950808877965961856, 2616769087480013696, 3515864044679266304, 4611826686121443328, 9223371897268338688

Offset: 1

Author

Robert G. Wilson v, Jul 26 2000

Keywords

• nonn

Comments

Or, numbers n such that sigma(n) = k*n + d(n) for some k.

For most terms > 4, sigma(n) = 2*n + d(n), i.e., k=2. However, for the 12th term, k=3.

If p = 2^m-(2m+1) is prime and n = 2^(m-1)*p then sigma(n) = 2*n+d(n), i.e., k=2 and n is in the sequence. 56, 7232, 30592, 36028789368553472, 9223371897268338688 and 29230032746618058364071726105239688547563879792624 are such terms of the sequence. - Farideh Firoozbakht, Aug 19 2013

Note that for m = a(17) = 47319950478240, we have (sigma(m) - d(m))/m = 4. - Max Alekseyev, May 31 2025

Links

• Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for perfect numbers, Journal of Integer Sequences 13.3 (2010), 18 pp. Article ID 10.3.1.

Crossrefs

Cf. A000203, A000005, A054024.

Programs

Mathematica

Do[If[Mod[DivisorSigma[1, n]-DivisorSigma[0, n], n]==0, Print[n]], {n, 1, 10^8}]

PARI

is(n)=my(f=factor(n)); (sigma(f)-numdiv(f))%n==0 \\ Charles R Greathouse IV, Nov 04 2016

Formula

Numbers n such that A000203(n) (mod n) == A000005(n) or A054024(n)=A000005(n). - Labos Elemer, Apr 12 2002

Extensions

a(15) from Giovanni Resta, Nov 07 2019

a(16)-a(29) from Max Alekseyev, May 31 2025