A215012 Composite numbers n such that sigma(n)/n leaves a remainder which divides n.

12, 18, 20, 24, 40, 56, 88, 104, 180, 196, 224, 234, 240, 360, 368, 420, 464, 540, 600, 650, 780, 992, 1080, 1344, 1504, 1872, 1888, 1890, 1952, 2016, 2184, 2352, 2376, 2688, 3192, 3276, 3724, 3744, 4284, 4320, 4680

Offset

1,1

Comments

The numbers and the program were provided by Charles R Greathouse IV.

If n belongs to the sequence, then sigma(n) = d*n + rem, so sigma(n)/n = d + rem/n. Since rem is a divisor of n, n = rem*r, thus rem/n = 1/r. Then sigma(n)/n = d + 1/r and contfrac(sigma(n)/n) = [d, r], and length(contfrac(sigma(n)/n)) = 2. That is, A071862(n) = 2. - Michel Marcus, Aug 29 2012

Links

Donovan Johnson, Table of n, a(n) for n = 1..1000

Example

24 has the divisors 1,2,3,4,6,12,24, which sum to be 60. Divide 60 by 24 and the remainder is 12, which is a divisor of 24.

Mathematica

a={}; For[n=1, n<=5000, n++, If[!PrimeQ[n], {s=DivisorSigma[1, n]; If[Mod[n, Mod[s, n]] == 0, AppendTo[a, n]]; }]; ]; a  (* John W. Layman, Jul 31 2012 *)
Select[Range[5000], CompositeQ[#]&&Mod[#, Mod[DivisorSigma[1, #], #]]==0&] // Quiet (* Harvey P. Dale, May 24 2019 *)

Prog

(PARI) is(n)=my(t=sigma(n)%n); t && n%t==0 && !isprime(n)

Crossrefs

Cf. A000203, A071862.

Sequence in context: A087245 A342100 A153501 * A181595 A263189 A263838

Adjacent sequences: A215009 A215010 A215011 * A215013 A215014 A215015

Keyword

nonn

Author

J. M. Bergot, Jul 31 2012

Extensions

Terms a(24)-a(41) from John W. Layman, Jul 31 2012

Status

approved