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Pandigital biperiod squares: pandigital squares whose digits repeat twice in order.
3

%I #15 Mar 24 2024 10:49:23

%S 183673469387755102041183673469387755102041,

%T 326530612244897959184326530612244897959184,

%U 734693877551020408164734693877551020408164,132231404958677685950413223140496132231404958677685950413223140496

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

%C 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.

%D 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.

%H Chai Wah Wu, <a href="/A181789/b181789.txt">Table of n, a(n) for n = 1..34</a>

%o (Python)

%o from itertools import count, islice

%o from sympy import sqrt_mod

%o def A181789_gen(): # generator of terms

%o for j in count(9):

%o b = 10**j

%o a = b*10+1

%o for k in sorted(sqrt_mod(0,a,all_roots=True)):

%o if a*b <= (m:=k**2) < a*(a-1) and len(set(str(m//a))) == 10:

%o yield m

%o A181789_list = list(islice(A181789_gen(),20)) # _Chai Wah Wu_, Mar 23 2024

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

%K nonn,base

%O 1,1

%A _William Rex Marshall_, Nov 12 2010