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Megaperfect numbers: numbers n where A019294(n) = min {m: n divides sigma^(m) (n)} increases to a record; sigma^(m) means apply the sum-of-divisors function m times.
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%I #25 Jan 10 2020 13:12:43

%S 1,2,3,5,9,11,23,25,29,59,67,101,131,173,202,239,353,389,401,461,659,

%T 1319,1579,1847,2309,2797

%N Megaperfect numbers: numbers n where A019294(n) = min {m: n divides sigma^(m) (n)} increases to a record; sigma^(m) means apply the sum-of-divisors function m times.

%C Where records occur in A019294. a(n>=23) depend on a few probable primes.

%H Graeme L. Cohen & Herman J. J. te Riele, <a href="http://web.archive.org/db.cwi.nl/rapporten/abstract.php?abstractnr=897">Iterating the Sum-of-Divisors Function</a>: Abstract. (Page no longer available; link gives latest snapshot on web.archive.org from Sept. 2006)

%H Graeme L. Cohen & Herman J. J. te Riele, <a href="http://repos.project.cwi.nl:8888/cwi_repository/docs/I/10/10355A.pdf">Iterating the Sum-of-Divisors Function</a> [Broken link to a file "10355A.pdf", maybe the same as NM-R9525.pdf available through the above page of abstract.]

%H Graeme L. Cohen and Herman J. J. te Riele, <a href="https://projecteuclid.org/euclid.em/1047565640">Iterating the sum-of-divisors function</a>, Experimental Mathematics, 5 (1996), pp. 93-100.

%t f[n_, m_] := Block[{d = DivisorSigma[1, n]}, If[Mod[d, m] == 0, 0, d]]; g[n_] := Length[ NestWhileList[ f[ #, n] &, n, # != 0 &]] - 1; a = 0; Do[b = g[n]; If[b > a, a = b; Print[ n]], {n, 460}] (* _Robert G. Wilson v_, Jun 24 2005 *)

%o (PARI) m=0;for(n=1,oo,m<(m=max(A019294(n),m))&&print1(n",")) \\ _M. F. Hasler_, Jan 07 2020

%Y Cf. A019277 (the record values), A019294 (min{m: n|sigma^(m)(n)}), A019295 (ratio sigma^(m)(n)/n).

%K hard,nonn

%O 1,2

%A _N. J. A. Sloane_