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Search: id:A051664
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| A051664 |
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Number of terms in n-th cyclotomic polynomial. |
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+0
10
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| 2, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 7, 2, 17, 3, 19, 5, 9, 11, 23, 3, 5, 13, 3, 7, 29, 7, 31, 2, 15, 17, 17, 3, 37, 19, 17, 5, 41, 9, 43, 11, 7, 23, 47, 3, 7, 5, 23, 13, 53, 3, 17, 7, 25, 29, 59, 7, 61, 31, 9, 2, 31, 15, 67, 17, 31, 17, 71, 3, 73, 37, 7, 19, 31, 17, 79, 5, 3
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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a(n)=p(n) if n=p(n); a(n) is not always A006530(n). - Labos E. (labos(AT)ana.sote.hu), May 03 2002
This sequence is the Mobius transform of A087073. Let m be the squarefree part of n, then a(n) = a(m). When n = pq, the product of two distinct odd primes, then there is a formula for a(pq). Let x = 1/p (mod q) and y = 1/q (mod p). Then a(pq) = 2xy-1. There are also formulas for the number of positive and negative terms. See papers by Carlitz or Lam and Leung. - T. D. Noe (noe(AT)sspectra.com), Aug 08 2003
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REFERENCES
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L. Carlitz, Number of terms in the cyclotomic polynomial F(pq,x), Amer. Math. Monthly, Vol. 73, No. 9, 1966, pp. 979-981.
T. Y. Lam and K. H. Leung, On the Cyclotomic Polynomial Phi(pq,x), Amer. Math. Monthly, Vol. 103, No. 7, 1996, pp. 562-564.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
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FORMULA
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a(n)=phi(n)+1-A086798(n) - T. D. Noe (noe(AT)sspectra.com), Aug 08 2003
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EXAMPLE
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9th cyclotomic polynomial is x^6+x^3+1 which has 3 terms, so a(9)=3.
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MATHEMATICA
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Table[Count[CoefficientList[Cyclotomic[n, x], x], _?(#!=0&)], {n, 0, 100}]
Table[Length[Cyclotomic[n, x]], {n, 1, 100}] - Artur Jasinski (grafix(AT)csl.pl), Jan 15 2007
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PROGRAM
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(PARI) a(n)=sum(k=0, eulerphi(n), if(polcoeff(polcyclo(n), k), 1, 0))
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CROSSREFS
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Cf. A086765 (number of positive terms in n-th cyclotomic polynomial), A086780 (number of negative terms in n-th cyclotomic polynomial), A086798 (number of zero terms in n-th cyclotomic polynomial), A087073.
Sequence in context: A151663 A162753 A111089 this_sequence A029656 A121306 A073311
Adjacent sequences: A051661 A051662 A051663 this_sequence A051665 A051666 A051667
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KEYWORD
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nonn
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AUTHOR
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Jud McCranie (j.mccranie(AT)comcast.net)
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EXTENSIONS
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More terms from Labos E. (labos(AT)ana.sote.hu), May 03 2002
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