

A153501


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


11



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
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OFFSET

1,1


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


LINKS

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


EXAMPLE

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


MAPLE

filter:= proc(n) local s; s:= numtheory:sigma(n); (s > 2*n) and (n mod (s2*n) = 0) end proc:
select(filter, [$1..10^5]); # Robert Israel, Nov 07 2014


PROG

(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


CROSSREFS

Intersection of A097498 and A005101.
Disjoint union of A181595 and A005820.
Cf. A000203, A033880.
Sequence in context: A290141 A087245 A342100 * A215012 A181595 A263189
Adjacent sequences: A153498 A153499 A153500 * A153502 A153503 A153504


KEYWORD

nonn


AUTHOR

Donovan Johnson, Jan 02 2009


STATUS

approved



