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A364369 a(n) is the least prime that is the concatenation of n squares, where the concatenations of the last k of these squares are prime for 2 <= k <= n.

%I A364369 #15 Oct 22 2023 01:25:24
%S A364369 11,419,4919,181919,1981919,49936919,991981919,9991981919,
%T A364369 16999369225919,136999369225919,99361981818199181,1729936999369225919,
%U A364369 3681225936999369225919,132481225936999369225919,99362500576936999369225919,8199362500576936999369225919
%N A364369 a(n) is the least prime that is the concatenation of n squares, where the concatenations of the last k of these squares are prime for 2 <= k <= n.
%C A364369 At most 2 of the n squares are not divisible by 9.
%e A364369 a(5) = 181919 because it is the concatenation of five squares 1^2 = 1, 9^2 = 81, 3^2 = 9, 1^2 = 1 and 3^2 = 9, and 19, 919, 81919 and 181919 are all prime, and this is the least number that works.
%p A364369 for d from 1 to 8 do
%p A364369   m1:= ceil(10^((d-1)/2));
%p A364369   m2:= floor(sqrt(10^d - 1));
%p A364369   S[d]:= {seq(i^2, i=m1..m2)};
%p A364369   if m1::even then m1:= m1+1 fi;
%p A364369   So[d]:= {seq(i^2, i=m1..m2,2)};
%p A364369 od:
%p A364369 for d from 2 to 9 do P[2,d]:= select(isprime, {seq(seq(seq(10^i*s+t, t=So[i]),s=S[d-i]),i=1..d-1)}) od:
%p A364369 L[2]:= 11:
%p A364369 mm[2]:= 2:
%p A364369 for m from 3 do
%p A364369    found:= false;
%p A364369    for d from mm[m-1]+1 to m+7 do
%p A364369      P[m,d]:= select(isprime, {seq(seq(seq(10^i*s+t, t=P[m-1,i]),s=S[d-i]),i=mm[m-1]..d-1)});
%p A364369      if P[m,d] <> {} and not found then mm[m]:= d; found:= true: L[m]:= min(P[m,d]) fi;
%p A364369    od;
%p A364369    if not found then break fi;
%p A364369 od:
%p A364369 seq(L[i],i=2..m-1);
%Y A364369 Cf. A167535, A364359.
%K A364369 nonn,base
%O A364369 2,1
%A A364369 _Robert Israel_, Oct 20 2023