A092118 Biperiod squares: square numbers whose digits repeat twice in order.

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

Table of n, a(n) for n=1..8.

Author?, MMB message board "big square"

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

Cf. A102567, A106497.

Sequence in context: A095438 A115503 A280299 * A257376 A037017 A337094

Adjacent sequences: A092115 A092116 A092117 * A092119 A092120 A092121

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