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Let sigma_m(n) be result of applying the sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m(n) = k*n; sequence gives log_2 of the (2,2)-perfect numbers.
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%I #28 Dec 27 2021 21:09:13

%S 1,2,4,6,12,16,18,30,60

%N Let sigma_m(n) be result of applying the sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m(n) = k*n; sequence gives log_2 of the (2,2)-perfect numbers.

%C Cohen and te Riele prove that any even (2,2)-perfect number (a "superperfect" number) must be of the form 2^(p-1) with 2^p-1 prime (Suryanarayana) and the converse also holds. Any odd superperfect number must be a perfect square (Kanold). Searches up to > 10^20 did not find any odd examples. - _Ralf Stephan_, Jan 16 2003

%C See also the Cohen-te Riele links under A019276.

%H Graeme L. Cohen and Herman 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.

%F Coincides with A000043(n) - 1 unless odd superperfect numbers exist.

%Y Cf. A019278, A019279.

%K nonn,more

%O 1,2

%A _N. J. A. Sloane_

%E a(8)-a(9) from _Jud McCranie_, Jun 01 2000