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Search: id:A001115
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| A001115 |
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Maximal number of pairwise relatively prime polynomials of degree n over GF(2).
(Formerly M0575 N0209)
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+0
1
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| 1, 2, 3, 4, 6, 9, 14, 23, 38, 64, 113, 200, 358, 653, 1202, 2223, 4151, 7781, 14659, 27721, 52603, 100084, 190969, 365134, 699617, 1342923, 2582172, 4972385, 9588933, 18515328, 35794987, 69278386, 134224480, 260309786, 505302925
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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For n>=4, a maximal set can be chosen by taking all irreducible polynomials of degree n, the squares of all irreducible polynomials of degree n/2 (if n is even) and, for each irreducible polynomial p of degree d with 1 <= d < n/2, a product p*q where q is irreducible of degree n-d. The q's should all be distinct, which is possible when n>=4.
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REFERENCES
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Bossen, D. C. and Yau, S. S.; Redundant residue polynomial codes. Information and Control 13 (1968) 597-618.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
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a(n) = P(n) + sum_{i from 1 to floor(n/2)} P(i), where P(n) = A001037(n) = number of irreducible polynomials of degree n.
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EXAMPLE
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n=1: x and x+1. n=2: x^2, x^2+1, x^2+x+1. n=3: x^3, x^3+1, x^3+x+1, x^3+x^2+1.
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MATHEMATICA
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p[0]=1; p[n_] := Sum[If[Mod[n, d]==0, MoebiusMu[n/d]2^d, 0], {d, 1, n}]/n; a[n_] := p[n]+Sum[p[i], {i, 1, Floor[n/2]}]
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CROSSREFS
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Sequence in context: A018140 A005579 A000381 this_sequence A096824 A089797 A081237
Adjacent sequences: A001112 A001113 A001114 this_sequence A001116 A001117 A001118
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Nov 18 2002
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