cp's OEIS Frontend

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.

%I A019285 #48 Dec 27 2021 21:09:27
%S A019285 60,240,960,4092,16368,58254,61440,65472,116508,466032,710400,983040,
%T A019285 1864128,3932160,4190208,67043328,119304192,268173312,1908867072,
%U A019285 7635468288,16106127360,711488165526,1098437885952,1422976331052
%N 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.
%C A019285 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
%C A019285 See also the Cohen-te Riele links under A019276.
%C A019285 No other terms < 5*10^11. - _Jud McCranie_, Feb 08 2012
%C A019285 1422976331052 is also a term. See comment in A019278. - _Michel Marcus_, May 15 2016
%C A019285 a(25) > 4*10^12. - _Giovanni Resta_, Feb 26 2020
%H A019285 G. L. Cohen and H. J. J. te Riele, <a href="http://projecteuclid.org/euclid.em/1047565640">Iterating the sum-of-divisors function</a>, Experimental Mathematics, 5 (1996), pp. 93-100.
%o A019285 (PARI) isok(n) = sigma(sigma(n))/n == 8; \\ _Michel Marcus_, May 15 2016
%Y A019285 Cf. A000668, A019276, A019278, A019279, A019281, A019282, A019283, A019284, A019285, A019286, A019287, A019288, A019289, A019290, A019291.
%K A019285 nonn,more
%O A019285 1,1
%A A019285 _N. J. A. Sloane_
%E A019285 a(19) from _Jud McCranie_, Nov 13 2001
%E A019285 a(20)-a(21) from _Jud McCranie_, Jan 29 2012
%E A019285 a(22)-a(24) from _Giovanni Resta_, Feb 26 2020