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A077440 Squares and their roots having square decimal digits.

%I A077440 #12 Aug 26 2024 13:45:06
%S A077440 0,1,100,10000,1000000,1100401,100000000,110040100,10000000000,
%T A077440 10100049001,11004010000,1000000000000,1010004900100,1100401000000,
%U A077440 100000000000000,100100004990001,101000490010000,110040100000000
%N A077440 Squares and their roots having square decimal digits.
%C A077440 If k is a term, then so is 100 * k. - _Robert Israel_, Aug 26 2024
%H A077440 Robert Israel, <a href="/A077440/b077440.txt">Table of n, a(n) for n = 1..388</a>
%e A077440 a(6) = 1100401 = 1049^2.
%e A077440 A019544(8)=441 is not a term, as 441=21^2 and 2 is not a square digit.
%p A077440 N:= 30: # for terms of up to 2*N digits
%p A077440 R:= {1}: T:= {1,9}:
%p A077440 for d from 2 to N do
%p A077440   T:= select(t -> convert(convert(t^2 mod 10^d, base,10),set) subset {0,1,4,9}, map(t -> (t, t + 10^(d-1), t + 4*10^(d-1), t + 9*10^(d-1)), T));
%p A077440   R:= R union select(t -> convert(convert(t^2,base,10),set) subset {0,1,4,9},T);
%p A077440 od:
%p A077440 R2:= map(t -> t^2, R):
%p A077440 Res:= map(t -> seq(t*10^(2*i), i=0..(2*N-ilog10(t)-1)/2), R2) union {0}:
%p A077440 sort(convert(Res,list)); # _Robert Israel_, Aug 26 2024
%t A077440 a = {}; Do[d = FromDigits[ ReplaceAll[ IntegerDigits[n, 4], {3 -> 9, 2 -> 4}]]; If[ Union[ Join[ IntegerDigits[d^2], {0, 1, 4, 9}]] == {0, 1, 4, 9}, a = Append[a, d^2]], {n, 0, 3*10^4}]; a
%Y A077440 a(n) = A077439(n)^2.
%Y A077440 Cf. A019544, A000290, A046030.
%K A077440 nonn,base
%O A077440 1,3
%A A077440 _Reinhard Zumkeller_, Nov 06 2002
%E A077440 Edited by _Robert G. Wilson v_, Nov 08 2002