A153501 Abundant numbers n such that n/(sigma(n)-2n) is an integer.

12, 18, 20, 24, 40, 56, 88, 104, 120, 196, 224, 234, 368, 464, 650, 672, 992, 1504, 1888, 1952, 3724, 5624, 9112, 11096, 13736, 15376, 15872, 16256, 17816, 24448, 28544, 30592, 32128, 77744, 98048, 122624, 128768, 130304, 174592, 396896, 507392

Offset: 1

Author

Donovan Johnson, Jan 02 2009

Keywords

• nonn

Comments

Sigma(n)-2n is the abundance of n.

The only odd term in this sequence < 2*10^12 is 173369889. - Donovan Johnson, Feb 15 2012

Equivalently, the abundancy of n, ab=sigma(n)/n, satisfies the following relation: numerator(ab) = 2*denominator(ab)+1, that is, ab=(2k+1)/k where k is the integer ratio mentioned in definition. - Michel Marcus, Nov 07 2014

The tri-perfect numbers (A005820) are in this sequence, since their abundancy is 3n/n = 3 = (2k+1)/k with k=1. - Michel Marcus, Nov 07 2014

Examples

			The abundance of 174592 = sigma(174592)-2*174592 = 43648. 174592/43648 = 4.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..200

Crossrefs

Intersection of A097498 and A005101.

Disjoint union of A181595 and A005820.

Cf. A000203, A033880.

Programs

Maple

filter:= proc(n) local s; s:= numtheory:-sigma(n); (s > 2*n) and (n mod (s-2*n) = 0) end proc:
select(filter, [$1..10^5]); # Robert Israel, Nov 07 2014

Mathematica

filterQ[n_] := Module[{s = DivisorSigma[1, n]}, s > 2n && Mod[n, s - 2n] == 0];
Select[Range[10^6], filterQ] (* Jean-François Alcover, Feb 01 2023, after Robert Israel *)

PARI

isok(n) = ((ab = (sigma(n)-2*n))>0) && (n % ab == 0) \\ Michel Marcus, Jul 16 2013

Sage

def A153501_list(len):
    def is_A153501(n):
        t = sigma(n,1) - 2*n
        return t > 0 and t.divides(n)
    return filter(is_A153501, range(1,len))
A153501_list(1000) # Peter Luschny, Nov 07 2014