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

1,1

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

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.

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

Cf. A092118, A102567.

Sequence in context: A370486 A116279 A369689 * A204097 A295355 A178221

Adjacent sequences: A106494 A106495 A106496 * A106498 A106499 A106500

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