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A073011
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Decimal expansion of Lehmer's constant (also known as the Salem constant).
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18
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1, 1, 7, 6, 2, 8, 0, 8, 1, 8, 2, 5, 9, 9, 1, 7, 5, 0, 6, 5, 4, 4, 0, 7, 0, 3, 3, 8, 4, 7, 4, 0, 3, 5, 0, 5, 0, 6, 9, 3, 4, 1, 5, 8, 0, 6, 5, 6, 4, 6, 9, 5, 2, 5, 9, 8, 3, 0, 1, 0, 6, 3, 4, 7, 0, 2, 9, 6, 8, 8, 3, 7, 6, 5, 4, 8, 5, 4, 9, 9, 6, 2, 0, 9, 6, 8, 3, 0, 1, 1, 5, 5, 8, 1, 8, 1, 5, 3, 9, 4, 6, 5, 9, 2, 0
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OFFSET
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1,3
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COMMENTS
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This number is algebraic of degree 10.
The Salem constant given here is the smallest known value of Mahler's measure M(f)=abs(a_d)*Product_{k=1..d}max(1,abs(b_k)) of a polynomial f(x)=Sum_{k=0..d}(a_k*x^k)=a_d*Product_{k=1..d}(x-b_k). The minimum occurs for Lehmer's polynomial (coefficients A070178) L(x)=x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1 with M(L)=1.1762808... - Hugo Pfoertner, Mar 12 2006
The Salem numbers were named after the Greek mathematician Raphaël Salem (1898-1963). - Amiram Eldar, May 01 2021
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 1..5000
David Boyd, Small Salem numbers, Duke Math. Journal, vol. 44, 1977, pp. 315-328.
Henri Cohen, Leonard Lewin, and Don Zagier. A sixteenth-order polylogarithm ladder, Experimental Mathematics 1.1 (1992): 25-34.
Eriko Hironaka, What is Lehmer's number?, Notices Amer. Math. Soc., 56 (No. 3, 2009), 374-375.
D. H. Lehmer, Factorization of certain cyclotomic functions, Annals of Math. vol. 34, 1933, pp. 461-479.
Douglas Lind, Lehmer's Problem for compact abelian groups, arXiv:math/0303279 [math.NT], 2003-2014.
Michael Mossinghoff, Lehmer's Problem Website.
Michael Mossinghoff, Small Salem Numbers.
Simon Plouffe, Salem Constant.
Raphaël Salem, Power series with integral coefficients, Duke mathematical journal, Vol. 12, No. 1 (1945), pp. 153-172.
Eric Weisstein's World of Mathematics, Salem Constants.
Eric Weisstein's World of Mathematics, Polylogarithm.
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FORMULA
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This is the largest real root of the polynomial x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1.
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EXAMPLE
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1.17628081825991750654407033847403505069341580656469...
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MATHEMATICA
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RealDigits[x/.FindRoot[x^10+x^9-Total[x^Range[3, 7]]+x+1==0, {x, 1, 2}, WorkingPrecision->120]][[1]] (* Harvey P. Dale, Sep 08 2011 *)
Root[ x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1, 2] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 05 2013 *)
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PROG
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(PARI) default(realprecision, 250); L(x)=x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1; solve(x=1.1, 1.2, L(x))
(PARI) polrootsreal(Pol([1, 1, 0, -1, -1, -1, -1, -1, 0, 1, 1]))[2] \\ Charles R Greathouse IV, Sep 03 2014
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CROSSREFS
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Cf. A070178 (Coefficients of Lehmer's polynomial).
Sequence in context: A276459 A181152 A244920 * A086312 A214280 A291081
Adjacent sequences: A073008 A073009 A073010 * A073012 A073013 A073014
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KEYWORD
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cons,nonn
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AUTHOR
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Robert G. Wilson v, Aug 03 2002
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EXTENSIONS
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Edited by N. J. A. Sloane, May 01 2012
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STATUS
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approved
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