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

36363636364, 45454545455, 54545454546, 63636363637, 72727272728, 81818181819, 90909090910, 428571428571428571429, 571428571428571428572, 714285714285714285715, 857142857142857142858

Offset: 1

Author

Lekraj Beedassy, May 04 2005

Keywords

• base
• nonn

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

Examples

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

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

• David W. Wilson, Table of n, a(n) for n = 1..1098
• 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.

Crossrefs

Cf. A092118, A102567.

Programs

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

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