A181791 -id:A181791 - cp's OEIS frontend

A181789 Pandigital biperiod squares: pandigital squares whose digits repeat twice in order.

183673469387755102041183673469387755102041, 326530612244897959184326530612244897959184, 734693877551020408164734693877551020408164, 132231404958677685950413223140496132231404958677685950413223140496

Offset: 1

William Rex Marshall, Nov 12 2010

Ondrejka asks in Problem 1130(b) (see reference) what the smallest biperiod square is in which the ten decimal digits occur equally often (an equipandigital biperiod square), but it remains unknown whether any such square even exists.

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.

Chai Wah Wu, Table of n, a(n) for n = 1..34

Cf. A092118 (biperiod squares), A181790, A181791.

from itertools import count, islice
from sympy import sqrt_mod
def A181789_gen(): # generator of terms
    for j in count(9):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(0,a,all_roots=True)):
            if a*b <= (m:=k**2) < a*(a-1) and len(set(str(m//a))) == 10:
                    yield m
A181789_list = list(islice(A181789_gen(),20)) # Chai Wah Wu, Mar 23 2024

A181790 Numbers k such that k concatenated with itself is a pandigital biperiod square.

Original entry on oeis.org

183673469387755102041, 326530612244897959184, 734693877551020408164, 132231404958677685950413223140496, 206611570247933884297520661157025, 297520661157024793388429752066116, 404958677685950413223140495867769

Offset: 1

William Rex Marshall, Nov 12 2010

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.

Chai Wah Wu, Table of n, a(n) for n = 1..34

Cf. A102567, A181789, A181791.

from itertools import count, islice
from sympy import sqrt_mod
def A181790_gen(): # generator of terms
    for j in count(9):
        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) and len(set(str(m:=k**2//a))) == 10:
                    yield m
A181790_list = list(islice(A181790_gen(),20)) # Chai Wah Wu, Mar 23 2024

cp's OEIS Frontend

A181789 Pandigital biperiod squares: pandigital squares whose digits repeat twice in order.

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