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A330816 Numbers that reach 1 under the iterations of the map k -> k/d(k) if d(k) | k, and k -> k otherwise, where d(k) is the number of divisors of k (A000005).

%I A330816 #15 Jan 20 2020 13:02:50
%S A330816 1,2,8,12,80,96,240,2240,3600,4032,20160,215040,268800,387072,435456,
%T A330816 725760,6350400,77414400,94058496,97542144,139345920,162570240,
%U A330816 278691840,365783040,452874240,457228800,5486745600,61931520000
%N A330816 Numbers that reach 1 under the iterations of the map k -> k/d(k) if d(k) | k, and k -> k otherwise, where d(k) is the number of divisors of k (A000005).
%C A330816 If k is a term then k/d(k) is a term.
%C A330816 The corresponding numbers of iterations to reach 1 are 0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, ...
%C A330816 From _David A. Corneth_, Jan 16 2020: (Start)
%C A330816 Similar to the first comment: every term > 1 is a multiple of an earlier term.
%C A330816 Conjecture: the sequence is finite; all 45 terms are in Corneth's a-file. Heuristic evidence: I took the first few terms C and made the Cartesian product with C and the 101-smooth numbers <= 10^8, seeing which were terms and removing duplicates. This process was repeated a few times until no more new terms were found. The largest number of divisors of any of these terms is < 10^6. (End)
%H A330816 David A. Corneth, <a href="/A330816/a330816.gp.txt">Conjectured full sequence</a>
%e A330816 12 is a term since 12/d(12) = 12/6 = 2 and 2/d(2) = 2/2 = 1.
%t A330816 f[n_] := If[Divisible[n, (d = DivisorSigma[0, n])], n/d, n]; Select[Range[10^6], FixedPoint[f, #] == 1 &]
%Y A330816 Subsequence of A033950.
%Y A330816 Cf. A000005, A002182, A174457, A274468.
%K A330816 nonn,more
%O A330816 1,2
%A A330816 _Amiram Eldar_, Jan 01 2020