A106497 - OEIS

%I #34 Feb 19 2024 14:48:00

%S 36363636364,45454545455,54545454546,63636363637,72727272728,

%T 81818181819,90909090910,428571428571428571429,571428571428571428572,

%U 714285714285714285715,857142857142857142858

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

%C For the corresponding numbers N see A102567.

%C 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 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 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 David W. Wilson, <a href="/A106497/b106497.txt">Table of n, a(n) for n = 1..1098</a>

%H Dr Barker, <a href="https://www.youtube.com/watch?v=c1peEN5Q-0c">Can Numbers Like These Be Square?</a>, YouTube video, 2023.

%H 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 63636363637 is in the sequence because 63636363637^2 = 4049586776940495867769 is 40495867769 written twice.

%o (Python)

%o from itertools import count, islice

%o from sympy import sqrt_mod

%o def A106497_gen(): # generator of terms

%o     for j in count(0):

%o         b = 10**j

%o         a = b*10+1

%o         for k in sorted(sqrt_mod(0,a,all_roots=True)):

%o             if a*b <= k**2 < a*(a-1):

%o                 yield k

%o A106497_list = list(islice(A106497_gen(),10)) # _Chai Wah Wu_, Feb 19 2024

%Y Cf. A092118, A102567.

%K base,nonn

%O 1,1

%A _Lekraj Beedassy_, May 04 2005

%E a(7) from _Klaus Brockhaus_, May 06 2005

%E More terms from _David W. Wilson_, Nov 05 2006

%E Reference and cross-references added by _William Rex Marshall_, Nov 12 2010