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A073011 Decimal expansion of Lehmer's constant (also known as the Salem constant). 18
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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
LINKS
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.
FORMULA
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.
EXAMPLE
1.17628081825991750654407033847403505069341580656469...
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Cf. A070178 (Coefficients of Lehmer's polynomial).
Sequence in context: A276459 A181152 A244920 * A086312 A370746 A214280
KEYWORD
cons,nonn
AUTHOR
Robert G. Wilson v, Aug 03 2002
EXTENSIONS
Edited by N. J. A. Sloane, May 01 2012
STATUS
approved

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)