OFFSET
2,1
COMMENTS
The identical strings must contain at least one nonzero digit, so that a(n) > 0. - Alonso del Arte, Jun 20 2018
In Bridy et al. it is shown how to construct an example (although not necessarily the least example) for each integer base n >= 2. - Jeffrey Shallit, Jun 14 2021
REFERENCES
Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, Experimental Math, 28 (2019), 428-439.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers", Revised Edition 1997, p. 189.
LINKS
Robert Israel, Table of n, a(n) for n = 2..99
Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, preprint arXiv:1707.03894 [math.NT], July 14 2017.
FORMULA
a(j*k^2-1) = j if k >= 2 and j is squarefree. - Robert Israel, May 19 2024
EXAMPLE
The first few squares in binary are 1, 100, 1001, 10000, 11001, 100100. Thus we see that 100100, which is 36 in decimal, the square of 6, is the first square which is the concatenation of two identical bit patterns, and therefore a(2) = 6.
MAPLE
f:= proc(b)
local d, F, x, t, j;
for d from 1 do
F:= select(t -> t[2]::odd, ifactors(1+b^d)[2]);
x:= mul(t[1], t=F);
if x >= b^d then next fi;
j:= ceil(sqrt(b^(d-1)/x));
if j^2*x < b^d then return j*sqrt(x*(1+b^d)) fi
od
end proc:
map(f, [$2..40]); # Robert Israel, May 19 2024
CROSSREFS
KEYWORD
base,nonn
AUTHOR
EXTENSIONS
Name slightly adjusted by Alonso del Arte, Jun 20 2018
STATUS
approved