

A230043


Numbers whose abundancy sigma(n)/n is a rational cube.


3



1, 8232, 32640, 265825, 3846879, 6517665, 14705145, 16926000, 31441920, 56471688, 146475000, 211421364, 277368000, 369022500, 662518050, 679568670, 968353620, 2166699360, 3091750900, 3755367252, 4122716598, 6536970000, 9740587500, 10066738500, 12423246290
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OFFSET

1,2


COMMENTS

All terms listed in the data section are deficient, but all 8multiperfect numbers (which are abundant...) also belong to this sequence.
As with A230538, it is possible to find larger numbers with same ratio sigma(n)/n, in some cases using perfect numbers A000396 (see a230043.txt link).  Michel Marcus, Oct 30 2013
One motivation for this sequence lies in the fact that n*sigma(n) is a square (A069070) if and only if sigma(n)/n is a rational square. But this does not hold for higher powers: If sigma(n)/n = (p/q)^k then n*sigma(n) = (pq)^k (n/q^k)^2.  M. F. Hasler, Nov 02 2013
In his post to NMBRTHRY, Michiel Kosters gives a 233digit number x such that sigma(x^3) is a cube. Actually this x^3 also belongs to the sequence, although there are no cubes in the current data. He has found many others such cubes that belong here, the smallest of which is 3590918978816938469301573291605^3, x having 31 digits, and x^3 92 digits. Is it possible to find the smallest such cube, or even a smaller one?  Michel Marcus, Jan 02 2014


LINKS



EXAMPLE

For n=8232, sigma(n)/n = 1000/343 = (10/7)^3.


MAPLE

isQcube := proc(r)
isA000578(numer(r)) and isA000578(denom(r)) ;
end proc: # see A000578 for isA000578()
isA230043 := proc(n)
abu := numtheory[sigma](n)/n ;
isQcube(abu) ;
end proc:
for n from 1 do
if isA230043(n) then
printf("%d, \n", n);
end if;


PROG

(PARI) is_A230043(n) = ispower(sigma(n)/n, 3);


CROSSREFS

Cf. A069070 (abundancy is a square).


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



