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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. 14

%I #43 Feb 27 2020 04:22:05

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

%N 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.

%C 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

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

%C No other terms < 5 * 10^11. - _Jud McCranie_, Feb 08 2012

%C 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

%C a(11) > 4*10^12. - _Giovanni Resta_, Feb 26 2020

%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.

%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>

%t Do[If[DivisorSigma[1, DivisorSigma[1, n]]==6n, Print[n]], {n, 6000000}] (* _Farideh Firoozbakht_, Dec 05 2005 *)

%o (PARI) isok(n) = sigma(sigma(n))/n == 6; \\ _Michel Marcus_, May 12 2016

%Y Cf. A000668, A019278, A019279, A019282.

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

%K nonn,more

%O 1,1

%A _N. J. A. Sloane_

%E a(10) by _Jud McCranie_, Feb 08 2012

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)