A230043 -id:A230043 - cp's OEIS frontend

A069070 Numbers n such that n*sigma(n) is a perfect square.

1, 40, 81, 135, 216, 224, 400, 819, 1372, 3240, 3744, 4650, 6318, 18144, 21700, 27930, 30240, 32400, 32760, 69312, 71148, 91694, 111132, 174592, 175500, 185220, 215472, 241395, 278318, 293907, 327600, 336675, 362700, 386232, 515450, 958737

Offset: 1

Benoit Cloitre, Apr 05 2002

Also n such that the squarefree part of n (A007913) equals the squarefree part of sigma(n), A355928.
Also n such that abundancy of n, sigma(n)/n is a rational square. - Michel Marcus, Oct 06 2013
See A230043, resp. A230538, for n whose abundancy is a rational cube, resp. fourth power. - M. F. Hasler, Nov 02 2013

Giovanni Resta, Table of n, a(n) for n = 1..1896 (terms < 4*10^12, first 500 terms from Donovan Johnson)

Cf. A000203, A000290, A007913, A064987, A355928.
Cf. A008848, A027687 (subsequences).
Cf. also A230043, A230538.
Positions of 0's in A355929.

Select[Range[1000000],IntegerQ[Sqrt[# DivisorSigma[1,#]]]&] (* Harvey P. Dale, Dec 24 2012 *)

for(n=1,1000000,if(issquare(n*sigma(n)),print1(n,",")))

isok(n) = issquare(sigma(n)/n); \\ Michel Marcus, Oct 06 2013

More terms from Rick L. Shepherd, Apr 07 2002

A230538 Numbers whose abundancy sigma(n)/n is a rational fourth power.

Original entry on oeis.org

1, 976250, 6276690, 6542085247225

Offset: 1

Michel Marcus, Oct 23 2013

Subsequence of A069070.
Note that there exist several other large numbers with the same abundancy as a(3), that is sigma(6276690)/6276690 = 19837440/6276690 = 256/81. For this, consider the two numbers 559625737239 (3^10*23*107*3851) and 1373356918809 (3^6*23*137*547*1093), both of which have sigma(n)/n = 128/81. As they are coprime to the perfect numbers, except 6, it suffices to multiply them by those terms of A000396 to get an abundancy of 2*128/81 = 256/81. The smallest of these is the 14-digit number 15669520642692. - Michel Marcus, Oct 29 2013
It is also possible to get higher powers for sigma(n)/n, for instance, 1024/243 = (4/3)^5 with n=1556619120, 4096/729 = (4/3)^6 with n=1526227435825092000, 279936/78125 = (6/5)^7 with n=553131046875000, 1679616/390625 = (6/5)^8 with n=15487669312500000. - Michel Marcus, Oct 30 2013
6542085247225 is a term. - Hiroaki Yamanouchi, Sep 22 2014
a(5) > 10^13. - Giovanni Resta, Jun 16 2015

			For n = 976250, sigma(n)/n = 2024352/976250 = 1296/625 = (6/5)^4.

Cf. A069070 (square), A230043 (cube).

isok(n) = ispower(sigma(n)/n, 4); \\ Michel Marcus, Oct 23 2013

a(4) from Giovanni Resta, Jun 14 2015

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A069070 Numbers n such that n*sigma(n) is a perfect square.

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A230538 Numbers whose abundancy sigma(n)/n is a rational fourth power.

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