A260709 - OEIS

%I #29 Jan 02 2023 12:30:51

%S 2,5,13,52,241,241,436,1009,1009,1009,2641,2641,8089,8089,8089,8089,

%T 18001,18001,53881,53881,53881,53881,87481,87481,87481,87481,87481,

%U 87481,117049,117049,515761,515761,515761,515761,515761,515761,1083289,1083289,1083289,1083289

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

%C A variant of A081650 which uses the remainder modulo k^2 instead of the congruence (mod k^2).

%C Suggested by _Don Reble_ and R. Israel and the original title of A081650.

%H 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 R. Israel in reply to Don Reble, <a href="http://list.seqfan.eu/oldermail/seqfan/2015-November/015643.html">A081650</a>, SeqFan list, Nov. 17, 2015

%t (* to get the sequence up to B *)

%t VQR=Table[Union[Mod[Range[(n^2)/2]^2,n^2]],{n,2,17}];

%t 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 (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 (MATLAB)

%o N = 2*10^8;  % to get all terms <= N

%o B = ones(1,N);

%o B([1:floor(sqrt(N))].^2) = 0;

%o m = 1;

%o while true

%o   nsq = ones(m^2,1);

%o   sqs = unique(mod([1:m^2/2].^2, m^2));

%o   sqs = [sqs(sqs > 0), m^2];

%o   nsq(sqs) = 0;

%o   S = nsq * ones(1,ceil(N/m^2));

%o   S = reshape(S,1,numel(S));

%o   B(S(1:N)>0) = 0;

%o   v = find(B,1,'first');

%o   if numel(v) == 0

%o     break

%o   end

%o   A(m) = v;

%o   m = m + 1;

%o end

%o A % _Robert Israel_, Nov 17 2015

%Y Cf. A081650.

%K nonn

%O 1,1

%A _M. F. Hasler_, Nov 17 2015