

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
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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 NagellLjunggren Problem: Powers with Repetitive Representations, Experimental Math, 28 (2019), 428439.
R. Ondrejka, Problem 1130: Biperiod Squares, Journal of Recreational Mathematics, Vol. 14:4 (198182), 299. Solution by F. H. Kierstead, Jr., JRM, Vol. 15:4 (198283), 311312.


LINKS



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*(a1):
yield k


CROSSREFS



KEYWORD

base,nonn


AUTHOR



EXTENSIONS



STATUS

approved



