A073011 -id:A073011 - cp's OEIS frontend

A316608 Decimal expansion of the eighth smallest known Salem number.

1, 2, 3, 2, 6, 1, 3, 5, 4, 8, 5, 9, 3, 1, 2, 1, 0, 0, 3, 9, 6, 2, 7, 3, 1, 6, 9, 4, 8, 0, 7, 9, 0, 9, 7, 9, 1, 4, 1, 1, 5, 7, 7, 3, 7, 1, 2, 0, 9, 8, 3, 1, 0, 4, 6, 7, 2, 9, 9, 1, 6, 5, 8, 2, 0, 5, 3, 8, 3, 9, 3, 5, 1, 3, 7, 3, 8, 9, 5, 7, 0, 3, 1, 3, 4, 9, 7, 8, 4, 8, 8, 5, 3, 0, 3, 2, 4, 9, 2, 2, 3, 3

Offset: 1

Jean-François Alcover, Jul 08 2018

			1.2326135485931210039627316948079097914115773712098310467299165820538...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 20.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3 ), A306079 (sigma4), A316605 (sigma5), A316606 (sigma6), A316607 (sigma7), A316609 (sigma9), A316610 (sigma10).

c1 = {1, -1, 0, 0, 0, -1, 1, 0, 0, -1, 1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2] - 1]).c2;
sigma8 = Root[p, x, 2];
RealDigits[sigma8, 10, 102][[1]]

polrootsreal(1 - x - x^5 + x^6 - x^9 + x^10 - x^11 + x^14 - x^15 - x^19 + x^20)[2] \\ Charles R Greathouse IV, Feb 11 2025

p = 1 - x - x^5 + x^6 - x^9 + x^10 - x^11 + x^14 - x^15 - x^19 + x^20.

A316609 Decimal expansion of the ninth smallest known Salem number.

Original entry on oeis.org

1, 2, 3, 5, 6, 6, 4, 5, 8, 0, 3, 8, 9, 7, 4, 7, 3, 0, 8, 1, 0, 5, 1, 6, 9, 3, 5, 1, 5, 3, 1, 2, 6, 3, 4, 7, 9, 7, 2, 3, 5, 1, 0, 0, 4, 2, 7, 4, 6, 2, 3, 9, 0, 7, 7, 6, 5, 0, 4, 3, 8, 0, 7, 7, 2, 0, 6, 3, 1, 1, 9, 2, 4, 9, 4, 0, 2, 7, 8, 1, 4, 9, 8, 8, 9, 1, 6, 6, 0, 7, 4, 0, 4, 2, 6, 3, 8, 5, 9, 2, 5

Offset: 1

Jean-François Alcover, Jul 08 2018

			1.2356645803897473081051693515312634797235100427462390776504380772063...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 22.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3 ), A306079 (sigma4), A316605 (sigma5), A316606 (sigma6), A316607 (sigma7), A316608 (sigma8), A316610 (sigma10).

c1 = {1, 0, -1, -1, 0, 0, 0, 1, 1, 0, -1, -1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2] - 1]).c2;
sigma9 = Root[p, x, 2];
RealDigits[sigma9, 10, 101][[1]]

polrootsreal(1 - x^2 - x^3 + x^7 + x^8 - x^10 - x^11 - x^12 + x^14 + x^15 - x^19 - x^20 + x^22)[2] \\ Charles R Greathouse IV, Feb 11 2025

p = 1 - x^2 - x^3 + x^7 + x^8 - x^10 - x^11 - x^12 + x^14 + x^15 - x^19 - x^20 + x^22.

A316610 Decimal expansion of the tenth smallest known Salem number.

Original entry on oeis.org

1, 2, 3, 6, 3, 1, 7, 9, 3, 1, 8, 0, 3, 2, 3, 0, 4, 8, 9, 8, 9, 9, 0, 9, 4, 8, 6, 9, 8, 0, 2, 0, 5, 4, 5, 5, 3, 9, 4, 4, 8, 1, 9, 2, 0, 8, 3, 6, 7, 8, 6, 9, 5, 6, 3, 7, 9, 4, 7, 5, 3, 7, 8, 4, 1, 1, 1, 8, 3, 6, 9, 9, 9, 5, 6, 7, 1, 4, 1, 5, 6, 3, 4, 2, 7, 2, 4, 3, 8, 5, 4, 3, 6, 5, 4, 0, 5, 5, 6, 1, 8, 8

Offset: 1

Jean-François Alcover, Jul 08 2018

			1.2363179318032304898990948698020545539448192083678695637947537841118...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 16.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3 ), A306079 (sigma4), A316605 (sigma5), A316606 (sigma6), A316607 (sigma7), A316608 (sigma8), A316609 (sigma9).

c1 = {1, -1, 0, 0, 0, 0, 0, 0, -1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2] - 1]).c2;
sigma10 = Root[p, x, 2];
RealDigits[sigma10, 10, 102][[1]]

polrootsreal(1 - x - x^8 - x^15 + x^16)[2] \\ Charles R Greathouse IV, Feb 11 2025

p = 1 - x - x^8 - x^15 + x^16.

A070178 Coefficients of Lehmer's polynomial.

Original entry on oeis.org

1, 1, 0, -1, -1, -1, -1, -1, 0, 1, 1

Offset: 0

N. J. A. Sloane, May 13 2002

Mahler's measure M(f) of a polynomial f is defined to be the absolute value of the product of those roots of f which lie outside the unit disk, multiplied by the absolute value of the coefficient of the leading term of f. Of all polynomials with integer coefficients, Lehmer's 10th degree polynomial produces the smallest known M(f), given in A073011. - Hugo Pfoertner, Mar 12 2006

			Polynomial is 1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10.

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 205.

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.
Charles L. Samuels, The infimum in the metric Mahler measure, arXiv:1408.4165 [math.NT], 2014 (see page 2).

Cf. A073011 (Mahler's measure of Lehmer's polynomial).

A167289 Signature sequence of the smallest Salem number of degree 18 (A219300).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 8, 1, 7, 6, 5, 4, 3, 9, 2, 8, 1, 7, 6, 5, 4, 10, 3, 9, 2, 8, 1, 7, 6, 5, 11, 4, 10, 3, 9, 2, 8, 1, 7, 6, 12, 5, 11, 4, 10, 3, 9, 2, 8, 1, 7, 13, 6, 12, 5, 11, 4, 10, 3, 9, 2, 8, 14, 1, 7, 13, 6, 12, 5, 11, 4, 10, 3, 9, 15

Offset: 1

Roger L. Bagula, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..5000
Michael Mossinghoff, Lehmer's Problem Website.
Michael Mossinghoff, Small Salem Numbers.
Eric Weisstein's World of Mathematics, Signature Sequence.
Index entries for sequences related to signature sequences

Cf. A073011, A147851, A219300.

a = {1, -1, 1, -1, 0, 0, -1, 1, -1};
b = Join[a, {1}, Reverse[a]];
p[x_] = Sum[b[[n]]*x^(n - 1), {n, 1, Length[b]}];
m = Root[p[x], 2];
Take[Transpose[Sort[Flatten[Table[{i + j*m, i}, {i, 25}, {j, 17}], 1], #1[[1]] < #2[[1]] &]][[2]], 95]

A142155 Expansion of x/( 1+x-x^2-x^4-x^5-x^6-x^7+x^9+x^10 ).

Original entry on oeis.org

1, -1, 2, -3, 6, -9, 17, -27, 48, -80, 139, -233, 402, -680, 1165, -1979, 3382, -5754, 9822, -16727, 28531, -48613, 82893, -141268, 240847, -410505, 699808, -1192838, 2033410, -3466085, 5908459, -10071512, 17168221, -29265017, 49885842, -85035890, 144953845, -247090156, 421194210

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 15 2008

limit_{n->infinity} a(n+1)/a(n) = -1.70461...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (-1,1,0,1,1,1,1,0,-1,-1).
E. W. Weisstein, Polylogarithm, MathWorld (see the remark concerning Li_17).

Cf. A073011.

Rest[CoefficientList[Series[x/(1+x-x^2-x^4-x^5-x^6-x^7+x^9+x^10),{x,0,50}],x]]  (* Harvey P. Dale, Mar 03 2011 *)

x='x+O('x^50); Vec(x/(1+x-x^2-x^4-x^5-x^6-x^7+x^9+x^10)) \\ G. C. Greubel, Mar 05 2017

Generating function g(x) = x/( 1+x-x^2-x^4-x^5-x^6-x^7+x^9+x^10 ) = 1/(x^10* p(1/x)) with p(x)= 1 +x -x^3 -x^4 -x^5 -x^6 -x^8 +x^9 +x^10.

Definition simplified by the Assoc. Eds. of the OEIS, Jun 30 2010

A177738 a(n) = floor( (x^n - x^(-n)) / (x - x^(-1)) ) where x = Pi-2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 17, 20, 23, 26, 30, 35, 40, 46, 52, 60, 69, 78, 90, 103, 117, 134, 153, 175, 199, 228, 260, 297, 339, 387, 442, 505, 576, 658, 751, 858, 979, 1118, 1277, 1457, 1664, 1900, 2169, 2476, 2826

Offset: 0

Roger L. Bagula, May 12 2010

nonn

The ratio a(n+1)/a(n) approaches Pi-2 as n approaches infinity, and is lower than even Salem polynomial expansions based on A073011.

The idea is the emulation of quadratic beta integer domains using a transcendental number base with a ratio below A073011.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A073011, A029826, A000796.

Clear[a, n, b]; b = Pi - 2; a[n_] = (b^n - b^(-n))/(b - b^(-1));
Table[Floor[a[n]], {n, 0, 50}]

Undefined terminology removed from the definition - The Assoc. Eds. of the OEIS, May 14 2010

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A316608 Decimal expansion of the eighth smallest known Salem number.

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A316609 Decimal expansion of the ninth smallest known Salem number.

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A316610 Decimal expansion of the tenth smallest known Salem number.

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A070178 Coefficients of Lehmer's polynomial.

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A167289 Signature sequence of the smallest Salem number of degree 18 (A219300).

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A142155 Expansion of x/( 1+x-x^2-x^4-x^5-x^6-x^7+x^9+x^10 ).

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A177738 a(n) = floor( (x^n - x^(-n)) / (x - x^(-1)) ) where x = Pi-2.

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