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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).
3
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)
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.
Sequence in context: A126192 A272720 A331459 * A066471 A001229 A298640
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Jan 01 2020
STATUS
approved