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%I #27 Jan 11 2020 15:32:04
%S 1,2,4,5,7,15,16,17,78,97,101,120,174,214,239,261,263,296,380,557,
%T 1287,1524,1722,1911,2023,2373
%N Records in A019294, number of iterations of the sigma function to reach a multiple of the starting value.
%C Original name: Let sigma_m(n) be the result of applying the sum-of-divisors function m times to n; let m(n) = min m such that n divides sigma_m (n); let k(n) = sigma_{m(n)}(n)/n; sequence gives k(n) for the megaperfect numbers n, where m(n) increases.
%C Records in A019294. a(n>=23) depend on a few probable primes.
%C See also the Cohen-te Riele links under A019276.
%C The original name mentioned the sequence of ratios k, i.e., A019295(A019276) = (1, 2, 5, 24, 168, 1834560, 6516224, 881280, ...), at present not listed in the OEIS. - _M. F. Hasler_, Jan 07 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.
%F a(n) = A019294(A019276(n)). - _M. F. Hasler_, Jan 07 2020
%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[ a]], {n, 460}] (* _Robert G. Wilson v_, Jun 24 2005 *)
%o (PARI) {M=0; for(n=1,oo, my(s=n,m=1); while((s=sigma(s))%n,m++); m>M&&print1(M=m,","))} \\ _M. F. Hasler_, Jan 07 2020
%Y Cf. A019276 (megaperfect numbers: where A019294 has records), A019294 (min m: n|sigma^m(n)), A019295 (sigma^m(n)/n with m = A019294).
%K hard,nonn
%O 1,2
%A _N. J. A. Sloane_
%E Definition corrected by _M. F. Hasler_, Jan 07 2020
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