A019285 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,8)-perfect numbers.

60, 240, 960, 4092, 16368, 58254, 61440, 65472, 116508, 466032, 710400, 983040, 1864128, 3932160, 4190208, 67043328, 119304192, 268173312, 1908867072, 7635468288, 16106127360, 711488165526, 1098437885952, 1422976331052

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• more

Comments

If 2^p-1 is a Mersenne prime greater than 3 then m = 15*2^(p-1) is in the sequence. Because sigma(sigma(m)) = sigma(15*2^(p-1)) = sigma(24*(2^p-1)) = 60*2^p = 8*(15*2^(p-1)) = 8*m. So for n>1 15/2*(A000668(n)+1) is in the sequence. 60, 240, 960, 61440, 983040, 3932160, 16106127360 and 1729382256910270464042 are such terms. - Farideh Firoozbakht, Dec 05 2005

See also the Cohen-te Riele links under A019276.

No other terms < 5*10^11. - Jud McCranie, Feb 08 2012

1422976331052 is also a term. See comment in A019278. - Michel Marcus, May 15 2016

a(25) > 4*10^12. - Giovanni Resta, Feb 26 2020

Links

• G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Crossrefs

Cf. A000668, A019276, A019278, A019279, A019281, A019282, A019283, A019284, A019285, A019286, A019287, A019288, A019289, A019290, A019291.

Programs

PARI

isok(n) = sigma(sigma(n))/n == 8; \\ Michel Marcus, May 15 2016

Extensions

a(19) from Jud McCranie, Nov 13 2001

a(20)-a(21) from Jud McCranie, Jan 29 2012

a(22)-a(24) from Giovanni Resta, Feb 26 2020