

A215012


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


2



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
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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



