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A084109 n is congruent to 1 (mod 4) and is not the sum of two squares. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

REFERENCES

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 56.

LINKS

Jean-Christophe Hervé, Table of n, a(n) for n = 1..1000

H. Ehlich, Determinantenabschätzungen für binäre Matrizen, Math. Z. 83 (1964) 123-132.

D. Raghavarao, Some aspects of weighing designs, Ann. Math. Stat. 31 (1960) 878-884.

EXAMPLE

a(1) = 3*7 = 21, a(2) = 3*11 = 33, a(3) = 3*19 = 57, a(14) = 3^3*7 = 189.

MAPLE

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)}:

sort(convert(S, list)); # Robert Israel, Oct 25 2015

MATHEMATICA

a[m_] := Complement[Range[1, m, 4], Union[Flatten[Table[j^2+k^2, {j, 1, Sqrt[m], 2}, {k, 0, Sqrt[m], 2}]]]]

PROG

(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

CROSSREFS

Cf. A000952, A003432, A003433, A022544, A001481.

Sequence in context: A189986 A190299 A280262 * A016105 A187073 A271101

Adjacent sequences:  A084106 A084107 A084108 * A084110 A084111 A084112

KEYWORD

easy,nonn

AUTHOR

William P. Orrick, Jun 18 2003

STATUS

approved

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Last modified October 15 07:56 EDT 2019. Contains 328026 sequences. (Running on oeis4.)