login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A081650 Least nonsquare whose remainder modulo k^2 is a square for all 0 < k <= n. 2
2, 5, 13, 73, 409, 801, 1584, 2241, 30601, 30601, 78409, 156825, 862416, 862416, 7929009, 28173825, 196668004, 196668004 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
See A260709 for the (maybe more natural) variant of squares (mod k^2) instead of remainders equal to a square. - M. F. Hasler, Nov 17 2015
REFERENCES
Mark A. Herkommer, Number Theory, A Programmer's Guide, McGraw-Hill, New York, 1999, page 315.
LINKS
EXAMPLE
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.
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
PROG
(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)))
From M. F. Hasler, Nov 17 2015: (Start)
(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.
(PARI) t=2; for(n=1, 50, print1(t=A081650(n, t), ", ")) \\ (End)
(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
CROSSREFS
Sequence in context: A128029 A215167 A349735 * A092262 A365326 A096280
KEYWORD
nonn,more
AUTHOR
Robert G. Wilson v, Mar 26 2003
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
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 05:16 EDT 2024. Contains 371264 sequences. (Running on oeis4.)