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A366106 Primes that are the concatenation of three squares in base 10.

%I A366106 #21 Oct 08 2023 09:30:19
%S A366106 101,109,149,191,199,401,409,419,449,491,499,911,919,941,991,1049,
%T A366106 1181,1259,1361,1481,1499,1601,1609,1619,1699,1811,1949,2549,2591,
%U A366106 3691,4049,4259,4481,4649,4909,4919,4999,6449,6491,8101,8111,8191,9049,9161,9181,9491,9649,9811,9949,10009,10091
%N A366106 Primes that are the concatenation of three squares in base 10.
%C A366106 The three squares need not be distinct.
%C A366106 At least one of the squares must be divisible by 9.
%C A366106 The first term that is a concatenation of three squares in two different ways is 14411, the concatenation of 1 = 1^2, 441 = 21^2 and 1 = 1^2 and also 144 = 12^2, 1 = 1^2 and 1 = 1^2.
%C A366106 The first term that is a concatenation of three squares in three different ways is 1961441, the concatenation of 196 = 14^2, 144 = 12^2 and 1 = 1^2, of 196, 1 and 441 = 21^2, and of 1, 961 = 31^2 and 441.
%H A366106 Robert Israel, <a href="/A366106/b366106.txt">Table of n, a(n) for n = 1..10000</a>
%e A366106 a(16) = 1049 is a term because it is the concatenation of 1 = 1^2, 0 = 0^2 and 49 = 7^2.
%p A366106 M:= 5: # for terms < 10^M
%p A366106 S:= {}:
%p A366106 for a from 1 while a^2 < 10^(M-2) do
%p A366106   x:= a^2; mx:= length(x);
%p A366106   for b from 0 while b^2 < 10^(M-1-mx) do
%p A366106     y:= b^2; my:= max(1,length(y));
%p A366106     for c from 0 while c^2 < 10^(M-mx-my) do
%p A366106       v:= parse(cat(x,y,c^2));
%p A366106       if isprime(v) then S:= S union {v} fi;
%p A366106 od od od:
%p A366106 sort(convert(S,list));
%t A366106 a[maxSquareIndex_Integer?Positive]:=Select[Flatten[Table[ToExpression[IntegerString[a^2]<>IntegerString[b^2]<>IntegerString[c^2]],{a,1,maxSquareIndex},{b,0,maxSquareIndex},{c,0,maxSquareIndex}]],PrimeQ]//Sort;a[10][[1;;51]] (* _Robert P. P. McKone_, Oct 02 2023 *)
%Y A366106 Cf. A167535.
%K A366106 nonn,base
%O A366106 1,1
%A A366106 _Robert Israel_, Sep 29 2023