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 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 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 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; end do: # R. J. Mathar, Oct 08 2013 PROG (PARI) is_A230043(n) = ispower(sigma(n)/n, 3); CROSSREFS Cf. A069070 (abundancy is a square). Sequence in context: A031844 A210008 A168632 * A045056 A221482 A237221 Adjacent sequences: A230040 A230041 A230042 * A230044 A230045 A230046 KEYWORD nonn AUTHOR Michel Marcus, Oct 06 2013 EXTENSIONS a(11)-a(25) from Donovan Johnson, Oct 10 2013 a(26)-a(33) from Donovan Johnson, Dec 22 2013 STATUS approved

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Last modified February 27 19:24 EST 2024. Contains 370378 sequences. (Running on oeis4.)