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Search: id:A159909
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| A159909 |
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Number of pairs (p,q) of odd primes p < q < r=prime(n) such that the cyclotomic polynomial Phi(p*q*r) has no coefficient > 1 in absolute value. |
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3
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| 0, 0, 0, 0, 1, 0, 1, 2, 3, 3, 4, 2, 7, 1, 3, 2, 6, 6, 4, 7, 9, 6, 5, 10, 7, 9, 8, 6, 13, 9, 4, 14, 10, 10, 18, 6, 12, 12, 10, 16, 15, 11, 18, 14, 11, 19
(list; graph; listen)
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OFFSET
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1,8
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COMMENT
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The cyclotomic polynomial Phi[pqr] can only have coefficients with absolute value > 1 if p,q,r are distinct odd primes, that's why we require 2 < p < q < r. If any of these inequalities is replaced by equality, then Phi[pqr] necessarily has only zero or unit (+-1) coefficients. Sequence A159908 counts all possibilities including these trivial cases.
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LINKS
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Phil Carmody, "Cyclotomic polynomial puzzles", in: "primenumbers" group, May 9, 2009.
Eric W. Weisstein, "Cyclotomic Polynomial", in: MathWorld--A Wolfram Web Resource.
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EXAMPLE
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a(5)=1 is the first nonzero term, since the smallest example for Phi(pqr) having no coefficient > 1 (in abs. value) for odd primes p<q<r is obtained for r=prime(5), namely Phi(3*7*11).
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PROGRAM
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(PARI) A159909(n) = sum( i=2, n-1, my(pq=prime(n)*prime(i)); sum( j=2, i-1, vecmax(abs(Vec(polcyclo(prime(j)*pq))))==1 ))
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CROSSREFS
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Cf. A117223 [From T. D. Noe (noe(AT)sspectra.com), May 11 2009]
Sequence in context: A060573 A103893 A106448 this_sequence A098007 A007554 A139069
Adjacent sequences: A159906 A159907 A159908 this_sequence A159910 A159911 A159912
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KEYWORD
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hard,more,nonn
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AUTHOR
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M. F. Hasler (MHasler(AT)univ-ag.fr), May 09 2009
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EXTENSIONS
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Extended by T. D. Noe (noe(AT)sspectra.com), May 11 2009
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