A230043 Numbers whose abundancy sigma(n)/n is a rational cube.

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

Offset: 1

Author

Michel Marcus, Oct 06 2013

Keywords

• nonn

Comments

All terms listed in the data section are deficient, but all 8-multiperfect 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 233-digit 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

Examples

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

Links

• Michel Marcus and Donovan Johnson, Table of n, a(n) for n = 1..33
• A. Flammenkamp, The multiply perfect numbers page
• M. Kosters, A solution to sigma(x^3)=y^3, Post to NMBRTHRY, Dec 23 2013
• Michel Marcus, Some other terms that belong to the sequence

Crossrefs

Cf. A069070 (abundancy is a square).

Programs

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;
end do: # R. J. Mathar, Oct 08 2013

PARI

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

Extensions

a(11)-a(25) from Donovan Johnson, Oct 10 2013

a(26)-a(33) from Donovan Johnson, Dec 22 2013