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A086811
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Average (scaled by a certain explicit factor) over all integers k of a_k(n), the n-th coefficient of the k-th cyclotomic polynomial.
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0
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0, 3, 6, 16, 45, 126, 224, 1344, 684, 1116, 4752, 23760, 56784, 286944, 164664, 281472, 2449224, 7371648, 27086400, 160392960, 49635936, 68277888, 1049956992, 6077306880, 1252224000, 3240801792, 2083408128, 4066530048, 35225729280, 142745587200, 717382656000, 6279166033920, 2442775449600, 2080906813440, 2251759104000
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OFFSET
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1,2
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COMMENTS
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When n is odd the n-th term is an integer. If n is even then twice the n-th term is an integer. Conjecturally (Y. Gallot) the n-th term is always an integer. For n <= 128 this has been verified numerically by Yves Gallot. It is also an unproved conjecture due to H. Möller (1970) that no term of this sequence is negative.
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LINKS
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Table of n, a(n) for n=1..35.
Tom M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math. 41 (1972), 281-293.
Tom M. Apostol, The resultant of the cyclotomic polynomials F_m(ax) and F_n(bx), Math. Comp. 29 (1975), 1-6.
Gennady Bachman, On the coefficients of cyclotomic polynomials, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 1991, 86 pp.
Gennady Bachman, On the coefficients of cyclotomic polynomials, Mem. Amer. Math. Soc. 106 (1993), no. 510, 80 pp.
Eckford Cohen, A class of arithmetic functions, Proc. Natl. Acad. Sci. USA 41 (1955), 939-944.
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), 173-188.
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
H. G. Gadiyar and R. Padma, Linking the circle and the sieve: Ramanujan-Fourier series, arXiv:math/0601574 [math.NT], 2006.
Emiliano Gagliardo, Le funzioni simmetriche semplici delle radici n-esime primitive dell'unità, Bollettino dell'Unione Matematica Italiana Serie 3, 8(3) (1953), 269-273.
Yves Gallot, Pieter Moree, and Huib Hommersom, Value distribution of cyclotomic polynomial coefficients, arXiv:0803.2483 [math.NT], 2008.
Yves Gallot, Pieter Moree, and Huib Hommersom, Value distribution of cyclotomic polynomial coefficients, Unif. Distrib. Theory 6 (2011), 177-206.
Sherry Gong, On a problem regarding coefficients of cyclotomic polynomials, J. Number Theory 129 (2009), 2924-2932.
Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.
G. S. Kazandzidis, On the cyclotomic polynomial: Coefficients, Bull. Soc. Math. Grèce (N.S.) 4A (1963), 1-11.
G. S. Kazandzidis, On the cyclotomic polynomials: Morphology-Estimates, Bull. Soc. Math. Grèce (N.S.) 4A (1963), 50-73.
D. H. Lehmer, Some properties of the cyclotomic polynomial, J. Math. Anal. Appl. 15 (1966), 105-117.
H. Möller, Über die i-ten Koeffizienten der Kreisteilungspolynome, Math. Ann. 188 (1970), 26-38.
Pieter Moree and Huib Hommersom, Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients, arXiv:math/0307352 [math.NT], 2003.
K. Motose, Ramanujan's sums and cyclotomic polynomials, Math. J. Okayama U. 47, no 1, (2005), Article 5.
C. A. Nicol, On restricted partitions and a generalization of the Euler phi number and the Moebius function, Proc. Natl. Acad. Sci. USA 39(9) (1953), 963-968.
C. A. Nicol and H. S. Vandiver, A von Sterneck arithmetical function and restricted partitions with respect to a modulus, Proc. Natl. Acad. Sci. USA 40(9) (1954), 825-835.
K. G. Ramanathan, Some applications of Ramanujan's trigonometrical sum C_m(n), Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69.
Srinivasa Ramanujan, On certain trigonometric sums and their applications in the theory of numbers, Trans. Camb. Phil. Soc. 22 (1918), 259-276.
Wikipedia, Cyclotomic polynomial.
Wikipedia, Ramanujan's sum.
Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.
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FORMULA
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Let M_k = k * Product_{prime p<=k} p. Let q be any prime > k. Then the k-th term (for k >= 2) is M_k * Sum_{d|M_k} ( a_d(k) + a_{d*q}(k) )/(2*d). The average of the k-th coefficient of the n-th cyclotomic polynomial is given by the k-th coefficient of this sequence divided by Zeta(2) * k * Product_{p<=k} (p+1). (Zeta(2) = Pi^2/6.) [See Section 8.3 in Moree and Hommerson (2003).]
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MAPLE
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with(numtheory):for k from 1 to 50 do; v := 1: w := 1:j := 1:z := 1:while ithprime(j)<=k do; v := v*ithprime(j); w := w*(1+1/ithprime(j)); z := z*(ithprime(j)+1); j := j+1; end do: v := v*k:z := z*k:q := ithprime(j):te := 0:for i from 1 to nops(divisors(v)) do; d := divisors(v)[i]; kl(x) := 1; for j from 1 to k do; if modp(d, j)=0 then kl(x) := taylor(kl(x)*(1-x^j)^mobius(d/j), x, k+1); end if; end do: te := te+coeff(kl(x), x, k)/d; kl(x) := 1; for j from 1 to k do; if modp(q*d, j)=0 then kl(x) := taylor(kl(x)*(1-x^j)^mobius(q*d/j), x, k+1); end if; end do: te := te+coeff(kl(x), x, k)/d; end do: zr := te/(2*w):print(k, zr*z):end do:
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CROSSREFS
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Cf. A013595, A013596, A054532, A054533, A054534, A054535.
Sequence in context: A202846 A107269 A333026 * A106361 A113040 A244036
Adjacent sequences: A086808 A086809 A086810 * A086812 A086813 A086814
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KEYWORD
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frac,nonn
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AUTHOR
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Pieter Moree (moree(AT)mpim-bonn.mpg.de), Aug 05 2003
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EXTENSIONS
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More terms from Petros Hadjicostas, Aug 01 2019 using the author's Maple program
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STATUS
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approved
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