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A106497
Numbers whose square is the concatenation of two identical numbers, i.e., of the form NN.
48
36363636364, 45454545455, 54545454546, 63636363637, 72727272728, 81818181819, 90909090910, 428571428571428571429, 571428571428571428572, 714285714285714285715, 857142857142857142858
OFFSET
1,1
COMMENTS
For the corresponding numbers N see A102567.
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
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.
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.
LINKS
Dr Barker, Can Numbers Like These Be Square?, YouTube video, 2023.
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.
EXAMPLE
63636363637 is in the sequence because 63636363637^2 = 4049586776940495867769 is 40495867769 written twice.
PROG
(Python)
from itertools import count, islice
from sympy import sqrt_mod
def A106497_gen(): # generator of terms
for j in count(0):
b = 10**j
a = b*10+1
for k in sorted(sqrt_mod(0, a, all_roots=True)):
if a*b <= k**2 < a*(a-1):
yield k
A106497_list = list(islice(A106497_gen(), 10)) # Chai Wah Wu, Feb 19 2024
CROSSREFS
Sequence in context: A370486 A116279 A369689 * A204097 A295355 A178221
KEYWORD
base,nonn
AUTHOR
Lekraj Beedassy, May 04 2005
EXTENSIONS
a(7) from Klaus Brockhaus, May 06 2005
More terms from David W. Wilson, Nov 05 2006
Reference and cross-references added by William Rex Marshall, Nov 12 2010
STATUS
approved