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

1, 976250, 6276690, 6542085247225

Offset: 1

Author

Michel Marcus, Oct 23 2013

Keywords

• nonn
• bref
• more

Comments

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

Examples

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

Crossrefs

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

Programs

PARI

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

Extensions

a(4) from Giovanni Resta, Jun 14 2015