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A377087
Number of cycles under iteration of the map sending a positive integer to the product of its leading base-n digit and the sum of the squares of its base-n digits.
3
0, 1, 0, 1, 2, 1, 2, 0, 3, 2, 5, 2, 4, 1, 3, 3, 3, 2, 3, 4, 1, 5, 3, 7, 2, 3, 3, 3, 4, 4, 5, 3, 6, 12, 2, 1, 3, 2, 6, 10, 4, 8, 6, 3, 4, 2, 3, 1, 3, 4, 9, 3, 2, 2, 5, 7, 4, 8, 7, 5, 6, 6, 6, 1, 8, 7, 4, 6, 6, 2, 5, 7, 5, 5, 4, 5, 3, 4, 3, 5, 2, 4, 7, 8, 3, 7, 7
OFFSET
2,5
COMMENTS
If b>=2 and a>=b^3 then E(a,2,b)<a. For each positive integer a, there is an positive integer m such that E^m(a,2,b)<b^3. (Fox et al., 2024, Lemma 4).
LINKS
N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.
EXAMPLE
In the decimal system all integers go to (1), (298), (46, 208, 136), (26, 80, 512, 150), or (33, 54, 205, 58, 445, 228, 144) under iteration of the map A376270, hence there are two fixed points and three cycles. Therefore a(10) = 3.
CROSSREFS
A193585 is the analog for happy numbers.
Sequence in context: A365385 A366788 A290537 * A272569 A344788 A372687
KEYWORD
nonn,base
AUTHOR
N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 15 2024
STATUS
approved