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A260709 Smallest nonsquare congruent to a square (mod k^2) for all k = 1..n.

%I A260709 #30 Dec 23 2024 14:53:44
%S A260709 2,5,13,52,241,241,436,1009,1009,1009,2641,2641,8089,8089,8089,8089,
%T A260709 18001,18001,53881,53881,53881,53881,87481,87481,87481,87481,87481,
%U A260709 87481,117049,117049,515761,515761,515761,515761,515761,515761,1083289,1083289,1083289,1083289
%N A260709 Smallest nonsquare congruent to a square (mod k^2) for all k = 1..n.
%C A260709 A variant of A081650 which uses the remainder modulo k^2 instead of the congruence (mod k^2).
%C A260709 Suggested by _Don Reble_ and R. Israel and the original title of A081650.
%H A260709 Robert Israel and Emmanuel Vantieghem, <a href="/A260709/b260709.txt">Table of n, a(n) for n = 1..81</a>[Terms 1 through 70 were computed by R. Israel: terms 71 through 82 by E. Vantieghem. Nov 23 2015]
%H A260709 R. Israel in reply to Don Reble, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2015-November/015643.html">A081650</a>, SeqFan list, Nov. 17, 2015
%t A260709 (* to get the sequence up to B *)
%t A260709 VQR=Table[Union[Mod[Range[(n^2)/2]^2,n^2]],{n,2,17}];
%t A260709 Print[2];k=1;m=2;While[k<B,k++;m--;flag=0;While[flag==0,Label[m$];m++;If[!IntegerQ[Sqrt[m]],j=1;While[j<k,j++;If[! MemberQ[VQR[[j-1]],Mod[m,j^2]],Goto[m$]]];If[j==k,Print[m];flag=1]]]](* _Emmanuel Vantieghem_, Nov 23 2013 *)
%o A260709 (PARI) t=2;for(n=1,90, for(m=t,9e9,issquare(m)&&next; for(k=1,n,issquare(Mod(m,k^2))||next(2)); print1(t=m,",");break))
%o A260709 (MATLAB)
%o A260709 N = 2*10^8;  % to get all terms <= N
%o A260709 B = ones(1,N);
%o A260709 B([1:floor(sqrt(N))].^2) = 0;
%o A260709 m = 1;
%o A260709 while true
%o A260709   nsq = ones(m^2,1);
%o A260709   sqs = unique(mod([1:m^2/2].^2, m^2));
%o A260709   sqs = [sqs(sqs > 0), m^2];
%o A260709   nsq(sqs) = 0;
%o A260709   S = nsq * ones(1,ceil(N/m^2));
%o A260709   S = reshape(S,1,numel(S));
%o A260709   B(S(1:N)>0) = 0;
%o A260709   v = find(B,1,'first');
%o A260709   if numel(v) == 0
%o A260709     break
%o A260709   end
%o A260709   A(m) = v;
%o A260709   m = m + 1;
%o A260709 end
%o A260709 A % _Robert Israel_, Nov 17 2015
%Y A260709 Cf. A081650.
%K A260709 nonn
%O A260709 1,1
%A A260709 _M. F. Hasler_, Nov 17 2015