A059047 -id:A059047 - cp's OEIS frontend

A059046 Numbers n such that sigma(n)-n divides n-1.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 77, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211

Offset: 1

Jud McCranie, Dec 18 2000

nonn

Primes and prime powers (A000961) satisfy this equation, but other numbers do also (A059047). This sequence is the union of A000961 and A059047. These are related to hyperperfect numbers (A034897) in the cited paper by te Riele. [Mentions this sequence]

			For x=77, sigma(77)=96, 96-77=19, which divides 77-1.

G. C. Greubel, Table of n, a(n) for n = 1..1000
JRM Antalan, JAB Dris, Some New Results On Even Almost Perfect Numbers Which Are Not Powers Of Two, arXiv preprint arXiv:1602.04248, 2016
H. J. J. te Riele, Rules for constructing hyperperfect numbers, Fibonacci Quarterly, 22(1)1984, 50-60. See equation (3), the set M*.

Cf. A059047, A000203, A000961, A034897.

[n : n in [2..1000] | (n-1) mod (SumOfDivisors(n)-n) eq 0 ]; /* N. J. A. Sloane, Dec 23 2006 */

Select[Range[2,250],Divisible[#-1,DivisorSigma[1,#]-#]&] (* Harvey P. Dale, Jan 18 2011 *)

is(n)=n>1 && (n-1)%(sigma(n)-n)==0 \\ Charles R Greathouse IV, Oct 21 2015

A306531 Composite numbers k such that the sum of their aliquot parts divides k-1.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 77, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 611, 625, 729, 841, 961, 1024, 1073, 1331, 1369, 1681, 1849, 2033, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5293, 5329, 6031, 6241

Offset: 1

Paolo P. Lava, Feb 22 2019

			Aliquot parts of 77 are 1, 7, 11 and 78/(1+7+11) = 76/19 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Union of A059047 and A246547.

Cf. A203966, A306532.

with(numtheory): P:=proc(n) if not isprime(n) and frac((n-1)/(sigma(n)-n))=0 then n; fi; end: seq(P(i),i=2..6241);

q[k_] := !PrimeQ[k] && Divisible[k-1, DivisorSigma[1, k]-k]; Select[Range[2, 6500], q] (* Amiram Eldar, Jul 26 2025 *)

isok(n) = (n!=1) && !isprime(n) && !frac((n-1)/(sigma(n)-n)); \\ Michel Marcus, Feb 28 2019

cp's OEIS Frontend

A059046 Numbers n such that sigma(n)-n divides n-1.

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