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

%I A020339 #30 May 20 2024 02:53:06
%S A020339 6,2,615,84,119973,4,3,23620,36363636364,6,24766945690,17928148,915,4,
%T A020339 86808207405692007605,6,130,10,2667,95530227420606,10623969116570,12,
%U A020339 5,343872950627253606,9,14,59239353339085,8130
%N A020339 a(n)^2 is the least square base-n doublet (base-n representation is the concatenation of 2 identical strings).
%C A020339 The identical strings must contain at least one nonzero digit, so that a(n) > 0. - _Alonso del Arte_, Jun 20 2018
%C A020339 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
%D A020339 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.
%D A020339 David Wells, "The Penguin Dictionary of Curious and Interesting Numbers", Revised Edition 1997, p. 189.
%H A020339 Robert Israel, <a href="/A020339/b020339.txt">Table of n, a(n) for n = 2..99</a>
%H A020339 Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, <a href="https://arxiv.org/abs/1707.03894">The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations</a>, preprint arXiv:1707.03894 [math.NT], July 14 2017.
%H A020339 A. Ottens, <a href="https://web.archive.org/web/20041013050849/http://rec-puzzles.org:80/sol.pl/arithmetic/digits/squares/three.digits">The arithmetic-digits-squares-three.digits problem</a>
%F A020339 a(j*k^2-1) = j if k >= 2 and j is squarefree. - _Robert Israel_, May 19 2024
%e A020339 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.
%p A020339 f:= proc(b)
%p A020339   local d,F,x,t,j;
%p A020339   for d from 1 do
%p A020339     F:= select(t -> t[2]::odd, ifactors(1+b^d)[2]);
%p A020339     x:= mul(t[1],t=F);
%p A020339     if x >= b^d then next fi;
%p A020339     j:= ceil(sqrt(b^(d-1)/x));
%p A020339     if j^2*x < b^d then return j*sqrt(x*(1+b^d)) fi
%p A020339   od
%p A020339 end proc:
%p A020339 map(f, [$2..40]); # _Robert Israel_, May 19 2024
%Y A020339 Cf. A020340, A054214, A054215, A054216, A030465, A030466, A030467.
%K A020339 base,nonn
%O A020339 2,1
%A A020339 _David W. Wilson_
%E A020339 Name slightly adjusted by _Alonso del Arte_, Jun 20 2018