A174457 -id:A174457 - cp's OEIS frontend

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).

1, 2, 8, 12, 80, 96, 240, 2240, 3600, 4032, 20160, 215040, 268800, 387072, 435456, 725760, 6350400, 77414400, 94058496, 97542144, 139345920, 162570240, 278691840, 365783040, 452874240, 457228800, 5486745600, 61931520000

Offset: 1

Amiram Eldar, Jan 01 2020

If k is a term then k/d(k) is a term.
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, ...
From David A. Corneth, Jan 16 2020: (Start)
Similar to the first comment: every term > 1 is a multiple of an earlier term.
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)

			12 is a term since 12/d(12) = 12/6 = 2 and 2/d(2) = 2/2 = 1.

David A. Corneth, Conjectured full sequence

Subsequence of A033950.

Cf. A000005, A002182, A174457, A274468.

f[n_] := If[Divisible[n, (d = DivisorSigma[0, n])], n/d, n]; Select[Range[10^6], FixedPoint[f, #] == 1 &]

cp's OEIS Frontend

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).

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