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

Author

J. M. Bergot, Jul 31 2012

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A000203, A071862.

Programs

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 *)

PARI

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

Extensions

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