A081650 Least nonsquare whose remainder modulo k^2 is a square for all 0 < k <= n.
2, 5, 13, 73, 409, 801, 1584, 2241, 30601, 30601, 78409, 156825, 862416, 862416, 7929009, 28173825, 196668004, 196668004
Offset: 1
Examples
a(3) = 13 because for (mod 1) (A000037) is the set of all nonsquares, for (mod 4) (A079896) is the set beginning {5, 8, 12, 13, 17, 20, 21, 24, 28, 29, ...} and for (mod 9) (A081642) is the set beginning {10, 13, 18, 19, 22, 27, 28, 31, 37, 40, ...}. The first element of the intersection of these three sets is 13.
References
- Mark A. Herkommer, Number Theory, A Programmer's Guide, McGraw-Hill, New York, 1999, page 315.
Crossrefs
Programs
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MATLAB
N = 10^8; % to get all terms <= N B = ones(1,N); B([1:floor(sqrt(N))].^2) = 0; m = 1; while true nsq = ones(m^2,1); nsq([1:m].^2)=0; S = nsq * ones(1,ceil(N/m^2)); S = reshape(S,1,numel(S)); B(S(1:N)>0) = 0; v = find(B,1,'first'); if numel(v) == 0 break end A(m) = v; m = m + 1; end A % Robert Israel, Nov 17 2015 -
Maple
M:= 0: for m from 2 while M < 15 do if (not issqr(m)) and andmap(issqr, [seq(m mod k^2, k=1..M+1)]) then A[M+1]:= m; for k from M+2 while issqr(m mod k^2) do A[k]:= m od: M:= k-1; fi od: seq(A[m],m=1..15); # Robert Israel, Nov 17 2015 -
PARI
t=2; for(n=1,50, for(m=t,10^9, if(issquare(m), next); f=0; for(k=1,n,if(!issquare(m % k^2),f=1;break)); if(!f,print1(m","); t=m; break)))
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PARI
A081650(n,t=2)=for(m=t,9e9,issquare(m)&&next; for(k=1,n,issquare(m%k^2)||next(2));return(m)) \\ The 2nd optional arg allows us to give a lower search limit, useful since a(n+1) >= a(n) by definition: see usage below. t=2;for(n=1,50, print1(t=A081650(n,t),",")) \\ M. F. Hasler, Nov 17 2015
Extensions
Edited by Ralf Stephan, Mar 27 2003
Definition corrected and original PARI code updated by M. F. Hasler, Nov 17 2015
a(16) to a(18) from Robert Israel, Nov 17 2015
Comments