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
CROSSREFS
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