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

1,2

Comments

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)

Links

Table of n, a(n) for n=1..28.

David A. Corneth, Conjectured full sequence

Example

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

Mathematica

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

Crossrefs

Subsequence of A033950.

Cf. A000005, A002182, A174457, A274468.

Sequence in context: A126192 A272720 A331459 * A066471 A001229 A298640

Adjacent sequences: A330813 A330814 A330815 * A330817 A330818 A330819

Keyword

nonn,more

Author

Amiram Eldar, Jan 01 2020

Status

approved