A019283 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 (2,6)-perfect numbers.

42, 84, 160, 336, 1344, 86016, 550095, 1376256, 5505024, 22548578304

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• more

Comments

If 2^p-1 is a Mersenne prime then m = 21*2^(p-1) is in the sequence. Because sigma(sigma(m)) = sigma(sigma(21*2^(p-1))) = sigma(32*(2^p-1)) = 63*2^p = 6*(21*2^(p-1)) = 6*m. So 21*(A000668+1)/2 is a subsequence of this sequence. This is the subsequence 42, 84, 336, 1344, 86016, 1376256, 5505024, 22548578304, 24211351596743786496, ... - 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

Any odd perfect numbers must occur in this sequence, as such numbers must be in the intersection of A000396 and A326051, that is, satisfy both sigma(n) = 2n and sigma(2n) = 6n, thus in combination they must satisfy sigma(sigma(n)) = 6n. Note that any odd perfect number should occur also in A326181. - Antti Karttunen, Jun 16 2019

a(11) > 4*10^12. - Giovanni Resta, Feb 26 2020

Links

• Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
• Index entries for sequences where any odd perfect numbers must occur

Crossrefs

Cf. A000668, A019278, A019279, A019282.

Cf. A000203, A000396, A005820, A051027, A326051, A326181.

Programs

Mathematica

Do[If[DivisorSigma[1, DivisorSigma[1, n]]==6n, Print[n]], {n, 6000000}] (* Farideh Firoozbakht, Dec 05 2005 *)

PARI

isok(n) = sigma(sigma(n))/n  == 6; \\ Michel Marcus, May 12 2016

Extensions

a(10) by Jud McCranie, Feb 08 2012