[seqfan] Numbers that are "generated" in every base
njasloane at gmail.com
Sat Jan 1 05:25:45 CET 2022
Dear Sequence Fans,
Kaprekar says that n is "generated by k in base b" if n = k + S_b(k), where
S_b(k) is the sum of the digits of k when k is written in base b. For
example in base 10, 101 has two generators, 91 and 100. 101 is the smallest
number with two generators in base 10. For more background see the paper
that Max Alekseyev and I just finished:
http://neilsloane.com/doc/colombian12302021.pdf (it is also on the arXiv
but this version is better). A "self-number in base b" has no generator.
All numbers here are nonnegative, by the way.
What I am writing about is A230624, the list of numbers that have a
generator in every base b >= 2. There are 90 known terms (the b-file is
from Lars Blomberg). It begins 0, 2, 10, 14, 22, 38, 62, 94, ... It seems
it is not known whether this sequence is infinite. This seems like a very
nice problem, if anyone is interested. I created subsidiary sequences
A349820 - A349823, hoping some structure would emerge, without much success.
By the way, it is easy to see that all terms must be even, and if the
number is n = 2t, once b is greater than t, n is generated by the base-b
single-digit number t. So we only need to find generators for bases 2
through n/2. See A349223 for certificates for the first few terms of
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