cp's OEIS Frontend

A106497 Numbers whose square is the concatenation of two identical numbers, i.e., of the form NN.

%I A106497 #34 Feb 19 2024 14:48:00
%S A106497 36363636364,45454545455,54545454546,63636363637,72727272728,
%T A106497 81818181819,90909090910,428571428571428571429,571428571428571428572,
%U A106497 714285714285714285715,857142857142857142858
%N A106497 Numbers whose square is the concatenation of two identical numbers, i.e., of the form NN.
%C A106497 For the corresponding numbers N see A102567.
%C A106497 Numbers of the form j*(10^d + 1)/k where 10^d + 1 == 0 (mod k^2) and k/sqrt(10) < j < k. - _David W. Wilson_, Nov 09 2006
%D A106497 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 A106497 R. Ondrejka, Problem 1130: Biperiod Squares, Journal of Recreational Mathematics, Vol. 14:4 (1981-82), 299. Solution by F. H. Kierstead, Jr., JRM, Vol. 15:4 (1982-83), 311-312.
%H A106497 David W. Wilson, <a href="/A106497/b106497.txt">Table of n, a(n) for n = 1..1098</a>
%H A106497 Dr Barker, <a href="https://www.youtube.com/watch?v=c1peEN5Q-0c">Can Numbers Like These Be Square?</a>, YouTube video, 2023.
%H A106497 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.
%e A106497 63636363637 is in the sequence because 63636363637^2 = 4049586776940495867769 is 40495867769 written twice.
%o A106497 (Python)
%o A106497 from itertools import count, islice
%o A106497 from sympy import sqrt_mod
%o A106497 def A106497_gen(): # generator of terms
%o A106497     for j in count(0):
%o A106497         b = 10**j
%o A106497         a = b*10+1
%o A106497         for k in sorted(sqrt_mod(0,a,all_roots=True)):
%o A106497             if a*b <= k**2 < a*(a-1):
%o A106497                 yield k
%o A106497 A106497_list = list(islice(A106497_gen(),10)) # _Chai Wah Wu_, Feb 19 2024
%Y A106497 Cf. A092118, A102567.
%K A106497 base,nonn
%O A106497 1,1
%A A106497 _Lekraj Beedassy_, May 04 2005
%E A106497 a(7) from _Klaus Brockhaus_, May 06 2005
%E A106497 More terms from _David W. Wilson_, Nov 05 2006
%E A106497 Reference and cross-references added by _William Rex Marshall_, Nov 12 2010