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A081650
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Least nonsquare whose remainder modulo k^2 is a square for all 0 < k <= n.
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2
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2, 5, 13, 73, 409, 801, 1584, 2241, 30601, 30601, 78409, 156825, 862416, 862416, 7929009, 28173825, 196668004, 196668004
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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Mark A. Herkommer, Number Theory, A Programmer's Guide, McGraw-Hill, New York, 1999, page 315.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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PROG
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(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)))
(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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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Definition corrected and original PARI code updated by M. F. Hasler, Nov 17 2015
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STATUS
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approved
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