A377086 Number of fixed points 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.

1, 2, 2, 1, 1, 4, 3, 4, 2, 3, 1, 3, 5, 2, 4, 4, 2, 4, 1, 3, 3, 3, 1, 5, 2, 3, 5, 4, 4, 9, 2, 1, 1, 4, 2, 6, 4, 1, 2, 5, 3, 6, 3, 1, 2, 3, 1, 7, 2, 1, 3, 3, 1, 5, 4, 6, 5, 4, 2, 8, 3, 2, 7, 3, 1, 4, 4, 2, 3, 6, 3, 9, 2, 3, 4, 9, 3, 7, 3, 2, 6, 5, 1, 7, 3, 3, 3

Offset

2,2

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).

Unlike the related sequence A193583, which contains only odd numbers, this sequence contains even numbers and odd numbers.

Links

Nathan Fox, Table of n, a(n) for n = 2..100

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) = 2.

Crossrefs

Cf. A376270, A376272, A377087, A377088.

A193583 is the analog for happy numbers.

Sequence in context: A128315 A198329 A123566 * A279863 A201757 A053390

Adjacent sequences: A377083 A377084 A377085 * A377087 A377088 A377089

Keyword

nonn,base

Author

N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 15 2024

Status

approved