cp's OEIS Frontend

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

%I A056075 #40 Jun 01 2025 10:04:22
%S A056075 1,4,56,7192,7232,7912,10792,17272,30592,114256,2154584,3428368,
%T A056075 44375136,89245784,2739393699744,36993585958528,47319950478240,
%U A056075 118122891971648,943226995376128,2737657760695168,5020331545072768,36028789368553472,40256362055287184,42381542060395136,950808877965961856,2616769087480013696,3515864044679266304,4611826686121443328,9223371897268338688
%N A056075 Numbers m such that m divides sigma(m) - d(m).
%C A056075 Or, numbers n such that sigma(n) = k*n + d(n) for some k.
%C A056075 For most terms > 4, sigma(n) = 2*n + d(n), i.e., k=2. However, for the 12th term, k=3.
%C A056075 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
%C A056075 Note that for m = a(17) = 47319950478240, we have (sigma(m) - d(m))/m = 4. - _Max Alekseyev_, May 31 2025
%H A056075 Farideh Firoozbakht and M. F. Hasler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Hasler/hasler2.html">Variations on Euclid's formula for perfect numbers</a>, Journal of Integer Sequences 13.3 (2010), 18 pp. Article ID 10.3.1.
%F A056075 Numbers n such that A000203(n) (mod n) == A000005(n) or A054024(n)=A000005(n). - _Labos Elemer_, Apr 12 2002
%t A056075 Do[If[Mod[DivisorSigma[1, n]-DivisorSigma[0, n], n]==0, Print[n]], {n, 1, 10^8}]
%o A056075 (PARI) is(n)=my(f=factor(n)); (sigma(f)-numdiv(f))%n==0 \\ _Charles R Greathouse IV_, Nov 04 2016
%Y A056075 Cf. A000203, A000005, A054024.
%K A056075 nonn
%O A056075 1,2
%A A056075 _Robert G. Wilson v_, Jul 26 2000
%E A056075 a(15) from _Giovanni Resta_, Nov 07 2019
%E A056075 a(16)-a(29) from _Max Alekseyev_, May 31 2025