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Search: id:A013594
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| A013594 |
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Smallest order of cyclotomic polynomial containing n or -n as a coefficient. |
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+0
11
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| 0, 105, 385, 1365, 1785, 2805, 3135, 6545, 6545, 10465, 10465, 10465, 10465, 10465, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 15015, 11305, 17255, 17255, 20615, 20615, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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This sequence is infinite - see the Lang reference.
An alternative version would start with 1 rather than 0.
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REFERENCES
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S. Lang, Algebra: 3rd edition, Addison-Wesley, 1993, p. 281.
Other references on cyclotomic polynomials:
Bateman-Pomerance-Vaughan, Colloq. Math. Soc. Janos Bolyai 34 (1984), 171-202.
Erdos-Vaughan, J London Math Soc 8 (1974), 393-400.
Maier, Prog. Math. 85 (Birkhaueser), 1990, 349-366.
Maier, Acta Arith. 64 (1993), 227-235.
Maier, Prog Math. 139 (Birkhaueser) 1996, 633-638.
Maier, Period. Math. Hungar. 43 (2001), 155-169.
Montgomery-Vaughan, Glasgow Math. J. 27 (1985), 143-159.
Vaughan, Michigan Math. J. 21 (1974), 289-295.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
R. C. Vaughan, Bounds for the coefficients of cyclotomic polynomials, Michigan Math. J. 21 (1974), 289-295 (1975).
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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EXAMPLE
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a(2)=105 because cyclotomic(105) contains "-2" as coefficient, but for n < 105 cyclotomic(n) does not contain 2 or -2.
x^105 - 1 = ( - 1 + x)(1 + x + x^2)(1 + x + x^2 + x^3 + x^4)(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)(1 - x + x^3 - x^4 + x^5 - x^7 + x^8)(1 - x + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12)(1 - x + x^5 - x^6 + x^7 - x^8 + x^10 - x^11 + x^12 - x^13 + x^14 - x^16 + x^17 - x^18 + x^19 - x^23 + x^24)(1 + x + x^2 - x^5 - x^6 - 2x^7 - x^8 - x^9 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 - x^20 - x^22 - x^24 - x^26 - x^28 + x^31 + x^32 + x^33 + x^34 + x^35 + x^36 - x^39 - x^40 - 2x^41 - x^42 - x^43 + x^46 + x^47 + x^48)
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MATHEMATICA
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Table[Position[Table[Max[Abs[Flatten[CoefficientList[Transpose[FactorList[x^i - 1]][[1]], x]]]], {i, 1, 10000}], j][[1]], {j, 1, 10}] - Ian Miller (ianxmiller(AT)gmail.com), Feb 25 2008
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PROGRAM
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(PARI) nm=6545; m=0; forstep(n=1, nm, 2, if(issquarefree(n), p=polcyclo(n); o=poldegree(p); for(k=0, o, a=abs(polcoeff(p, k)); if(a>m, m=a; print([m, n, factor(n)])))))
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CROSSREFS
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Cf. A046887, A013595, A013596, A063696, A063698, A134518
Cf. A137979.
Sequence in context: A010090 A113480 A102792 this_sequence A136418 A134518 A078420
Adjacent sequences: A013591 A013592 A013593 this_sequence A013595 A013596 A013597
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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EXTENSIONS
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More terms from Eric Weisstein (eric(AT)weisstein.com)
Further terms from T. D. Noe, Oct 29 2007
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