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 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
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 (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 *)
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
KEYWORD
nonn
AUTHOR
Donovan Johnson, Jan 02 2009
STATUS
approved