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A344570 Number of pairs of n-digit squares such that the final (n-1) digits of the first square coincide with the initial (n-1) digits of the second.

%I A344570 #39 Jun 02 2021 03:04:59
%S A344570 0,4,6,7,10,5,10,4,15,0,13,5,16,3,57,2,8,2,5,1,119,1,13,8,39,5,55,2,
%T A344570 53,7,12,7,76,1,193,2,21,2,59,11,35,15,42,7,541,7,17,20,37,1,233,3,32,
%U A344570 2,373,19,65,0,15,16,181,15,8637,5,175,15
%N A344570 Number of pairs of n-digit squares such that the final (n-1) digits of the first square coincide with the initial (n-1) digits of the second.
%C A344570 For n=2, certain numbers (16 and 64) appear in more than one pair. No such numbers have been observed up to n=9, but so far there is no proof of this property.
%C A344570 a(10)=0 was found by Andrew Howroyd. Does this sequence contain infinitely many nonzero terms?
%C A344570 The sequence is inspired by a problem from the 2020 Polish Juniors' Mathematics Olympiad. It is problem 1: 'Is there a six-digit positive integer such that any two consecutive digits form a perfect square?'
%C A344570 Are there any other terms such that a(n) = 0 besides n=1 and n=10? - _Chai Wah Wu_, May 26 2021
%H A344570 Stowarzyszenie na rzecz Edukacji Matematycznej, <a href="https://omj.edu.pl/uploads/attachments/1etap20.pdf">Olimpiada Matematyczna Juniorów 2020/2021, etap 1</a> (in Polish).
%H A344570 Chai Wah Wu, <a href="/A344570/a344570_1.txt">pairs of squares for n = 1..46</a>
%H A344570 <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%e A344570 For n=2: (81,16), (16,64), (36,64), (64,49).
%e A344570 For n=3: (144,441), (196,961), (225,256), (625,256), (484,841), (784,841).
%e A344570 For n=4: (3136,1369), (4624,6241), (5184,1849), (5476,4761), (7396,3969), (7921,9216), (9409,4096).
%e A344570 For n=20: (64764644930975528100, 47646449309755281001) is the only pair. - _Andrew Howroyd_, May 23 2021
%o A344570 (PARI) a(n)={sum(k=sqrtint(10^(n-1))+1, sqrtint(10^n-1), my(t=k^2*10%10^n); t>=10^(n-1) && sqrtint(t+9)^2\10==t\10)} \\ _Andrew Howroyd_, May 24 2021
%Y A344570 Cf. A343855.
%K A344570 nonn,base,more
%O A344570 1,2
%A A344570 _Igor Trujnara_, May 23 2021
%E A344570 a(10)-a(20) from _Andrew Howroyd_, May 24 2021
%E A344570 a(21)-a(46) from _Chai Wah Wu_, May 26 2021
%E A344570 a(47)-a(66) from _Chai Wah Wu_, Jun 02 2021