A020339 a(n)^2 is the least square base-n doublet (base-n representation is the concatenation of 2 identical strings).

6, 2, 615, 84, 119973, 4, 3, 23620, 36363636364, 6, 24766945690, 17928148, 915, 4, 86808207405692007605, 6, 130, 10, 2667, 95530227420606, 10623969116570, 12, 5, 343872950627253606, 9, 14, 59239353339085, 8130

Offset: 2

Author

David W. Wilson

Keywords

• base
• nonn

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

Examples

			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.
		

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.
• A. Ottens, The arithmetic-digits-squares-three.digits problem

Crossrefs

Cf. A020340, A054214, A054215, A054216, A030465, A030466, A030467.

Programs

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

Formula

a(j*k^2-1) = j if k >= 2 and j is squarefree. - Robert Israel, May 19 2024

Extensions

Name slightly adjusted by Alonso del Arte, Jun 20 2018