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A092118
Biperiod squares: square numbers whose digits repeat twice in order.
3
1322314049613223140496, 2066115702520661157025, 2975206611629752066116, 4049586776940495867769, 5289256198452892561984, 6694214876166942148761, 8264462810082644628100, 183673469387755102041183673469387755102041
OFFSET
1,1
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.
MAPLE
f:=proc(n) local i, j, k; i:=cat(n, n); j:=convert(i, decimal, 10); issqr(j); end;
with(numtheory): Digits:=50:for d from 1 to 22 do tendp1:=10^d+1: tendp1fact:=ifactors(tendp1)[2]: n:=mul(piecewise(tendp1fact[i][2] mod 2=1, tendp1fact[i][1], 1), i=1..nops(tendp1fact)):for i from ceil(sqrt((10^(d-1))/n)) to floor(sqrt((10^d-1)/n)) do printf("%d, ", tendp1*n*i^2) od: od: # C. Ronaldo
PROG
(Python)
from itertools import count, islice
from sympy import sqrt_mod
def A092118_gen(): # generator of terms
for j in count(0):
b = 10**j
a = b*10+1
ab, aa = a*b, a*(a-1)
for k in sorted(sqrt_mod(0, a, all_roots=True)):
if ab <= (m:=k**2) < aa:
yield m
A092118_list = list(islice(A092118_gen(), 10)) # Chai Wah Wu, Mar 06 2024
CROSSREFS
Sequence in context: A095438 A115503 A280299 * A257376 A037017 A337094
KEYWORD
base,nonn
AUTHOR
Michael Mark, Dec 15 2004
EXTENSIONS
Corrected and extended by C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 15 2005
Definition corrected and improved, reference and cross-reference added by William Rex Marshall, Nov 12 2010
Keyword base added by William Rex Marshall, Nov 12 2010
STATUS
approved