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A084109
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n is congruent to 1 (mod 4) and is not the sum of two squares.
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8
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21, 33, 57, 69, 77, 93, 105, 129, 133, 141, 161, 165, 177, 189, 201, 209, 213, 217, 237, 249, 253, 273, 285, 297, 301, 309, 321, 329, 341, 345, 357, 381, 385, 393, 413, 417, 429, 437, 453, 465, 469, 473, 489, 497
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OFFSET
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1,1
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COMMENTS
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Alternatively, n is congruent to 1 (mod 4) with at least 2 distinct prime factors congruent to 3 (mod 4) in the squarefree part of n. - Comment corrected by Jean-Christophe Hervé, Oct 25 2015
Applications to the theory of optimal weighing designs and maximal determinants: An (n+1) X (n+1) conference matrix is impossible.
The upper bound of Ehlich/Wojtas on the determinant of a (0,1) matrix of order congruent to 1 (mod 4) cannot be achieved for n X n matrices.
The bound of Ehlich/Wojtas on the determinant of a (-1,1) matrix of order congruent to 2 (mod 4) cannot be achieved for (n+1) X (n+1) matrices.
Numbers with only odd prime factors, of which a strictly positive even number are raised to an odd power and congruent to 3 (mod 4). - Jean-Christophe Hervé, Oct 24 2015
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REFERENCES
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F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 56.
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LINKS
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EXAMPLE
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a(1) = 3*7 = 21, a(2) = 3*11 = 33, a(3) = 3*19 = 57, a(14) = 3^3*7 = 189.
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MAPLE
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N:= 1000: # to get all entries <= N
S:= {seq(i, i=1..N, 4)} minus
{seq(seq(i^2+j^2, j=1..floor(sqrt(N-i^2)), 2), i=0..floor(sqrt(N)), 2)}:
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MATHEMATICA
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a[m_] := Complement[Range[1, m, 4], Union[Flatten[Table[j^2+k^2, {j, 1, Sqrt[m], 2}, {k, 0, Sqrt[m], 2}]]]]
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PROG
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(PARI) is(n)=if(n%4!=1, return(0)); my(f=factor(n)); for(i=1, #f~, if(f[i, 1]%4==3 && f[i, 2]%2, return(1))); 0 \\ Charles R Greathouse IV, Jul 01 2016
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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