A019276 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.

1, 2, 3, 5, 9, 11, 23, 25, 29, 59, 67, 101, 131, 173, 202, 239, 353, 389, 401, 461, 659, 1319, 1579, 1847, 2309, 2797

Offset: 1

Author

N. J. A. Sloane

Keywords

• hard
• nonn

Comments

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

Links

• Graeme L. Cohen & Herman J. J. te Riele, Iterating the Sum-of-Divisors Function: Abstract. (Page no longer available; link gives latest snapshot on web.archive.org from Sept. 2006)
• Graeme L. Cohen & Herman J. J. te Riele, Iterating the Sum-of-Divisors Function [Broken link to a file "10355A.pdf", maybe the same as NM-R9525.pdf available through the above page of abstract.]
• Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Crossrefs

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

Programs

Mathematica

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 *)

PARI

m=0;for(n=1,oo,m<(m=max(A019294(n),m))&&print1(n",")) \\ M. F. Hasler, Jan 07 2020