A073011 -id:A073011 - cp's OEIS frontend

A261406 Numbers n such that the norm of Phi_n(alpha), the n-th cyclotomic polynomial evaluated at alpha, is +-1, where alpha is the Salem number defined in A073011.

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 18, 20, 21, 23, 24, 27, 28, 29, 30, 34, 37, 38, 40, 42, 44, 45, 47, 50, 52, 56, 57, 60, 63, 65, 66, 70, 74, 75, 76, 78, 84, 86, 92, 96, 98, 105, 110, 118, 132, 138, 144, 154, 160, 165, 186, 195, 204, 212, 240, 270, 286, 360

Offset: 1

N. J. A. Sloane, Aug 22 2015

nonn

Cohen, Henri, Leonard Lewin, and Don Zagier. "A sixteenth-order polylogarithm ladder." Experimental Mathematics 1.1 (1992): 25-34.

Cf. A073011.

A261407 Numbers d_m arising in Cohen et al.'s investigation of "ladder relations" among powers of the Salem number A073011.

Original entry on oeis.org

1, 2, 1, 24, 1, 81, 1, 63057, 1, 234280024, 1

Offset: 2

N. J. A. Sloane, Aug 22 2015

See Cohen et al. (1992) for precise definition.

Cohen, Henri, Leonard Lewin, and Don Zagier. "A sixteenth-order polylogarithm ladder." Experimental Mathematics 1.1 (1992): 25-34. See Table 1.

Cf. A073011.

A029826 Expansion of 1/(x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1) (inverse of Salem polynomial).

Original entry on oeis.org

1, -1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 3, 2, 4, 3, 5, 5, 6, 8, 8, 10, 12, 14, 16, 20, 22, 27, 31, 37, 44, 50, 61, 70, 83, 98, 115, 135, 159, 187, 220, 259, 304, 359, 420, 496, 583, 685, 807, 948, 1116, 1312, 1544

Offset: 0

Simon Plouffe

The root 1.1762808182599175065440703384740350... is the smallest known Salem number (A073011).

Simon Plouffe, Table of n, a(n) for n = 0..2998
Leonard Lewin, Structural Properties of Polylogarithms, AMS #37. p. 365, 1991.
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1,1,1,1,0,-1,-1).

Cf. A073011, A125950.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1))); // G. C. Greubel, May 07 2018

LinearRecurrence[{-1,0,1,1,1,1,1,0,-1,-1}, {1,-1,1,0,0,1,0,1,0,1}, 100] (* G. C. Greubel, May 07 2018 *)

Vec(1/(x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1)+O(x^66)) \\ Joerg Arndt, May 01 2018

a(n) = -a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7) - a(n-9) - a(n-10). - Roger L. Bagula and Gary W. Adamson, Oct 23 2008

A181600 Expansion of 1/(1 - x - x^2 + x^8 - x^10).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 85, 136, 218, 349, 559, 895, 1434, 2297, 3679, 5893, 9439, 15119, 24217, 38790, 62132, 99520, 159407, 255331, 408978, 655083, 1049283, 1680695, 2692063, 4312028, 6906816, 11063033, 17720278, 28383559, 45463532, 72821479

Offset: 0

Roger L. Bagula, May 06 2013

Limiting ratio is 1.60176..., the largest real root of -1 + x^2 - x^8 - x^9 + x^10. Compare this constant to Lehmer's Salem constant A073011 and the golden mean.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,0,0,0,-1,0,1).

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175773, A175782, A204631, A225391, A225393, A225394, A225482, A225499.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 -x-x^2+x^8-x^10))); // G. C. Greubel, Nov 03 2018

CoefficientList[Series[1/(1 - x - x^2 + x^8 - x^10), {x, 0, 50}], x]
LinearRecurrence[{1, 1, 0, 0, 0, 0, 0, -1, 0, 1}, {1, 1, 2, 3, 5, 8, 13, 21, 33, 53}, 50] (* Harvey P. Dale, Aug 11 2015 *)

Vec(1/(1 -x -x^2 +x^8 -x^10) + O(x^50)) \\ G. C. Greubel, Nov 16 2016

a(n) = a(n-1) + a(n-2) - a(n-8) + a(n-10). - Franck Maminirina Ramaharo, Oct 31 2018

A219300 Decimal expansion of the second smallest known Salem number.

Original entry on oeis.org

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

Offset: 1

Arkadiusz Wesolowski, Nov 17 2012

This number is algebraic of degree 18. Sequence A073011 contains the smallest known Salem number.

It is the only root r of the polynomial x^18 - x^17 + x^16 - x^15 - x^12 + x^11 - x^10 + x^9 - x^8 + x^7 - x^6 - x^3 + x^2 - x + 1 with abs(r) > 1. - Joerg Arndt, Nov 18 2012

			1.188368147508223588142960958....

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
M. J. Mossinghoff, Small Salem Numbers
Wikipedia, Salem number
Index entries for algebraic numbers, degree 18

Cf. A073011, A173911.

RealDigits[x /. FindRoot[x^18 - x^17 + x^16 - x^15 - x^12 + x^11 - x^10 + x^9 - x^8 + x^7 - x^6 - x^3 + x^2 - x + 1 == 0, {x, 1, 2}, WorkingPrecision -> 200]][[1]]

A306078 Decimal expansion of the third smallest known Salem number.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 19 2018

			1.200026523987391518902962100414601567240618151999851067924399839886...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 14.

Cf. A073011 (sigma1), A219300 (sigma2), A306079 (sigma4).

c1 = {1, 0, 0, -1, -1, 0, 0, 1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2]-1]). c2;
sigma3 = Root[p, x, 2];
RealDigits[sigma3, 10, 100][[1]]

polrootsreal(x^14 - x^11 - x^10 + x^7 - x^4 - x^3 + 1)[2] \\ Charles R Greathouse IV, Feb 11 2025

A306079 Decimal expansion of the fourth smallest known Salem number.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 19 2018

			1.20261674368860426111829541594861904534394983496952304368530957672645...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 14.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3).

c1 = {1, 0, -1, 0, 0, 0, 0, -1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2]-1]).c2;
sigma4 = Root[p, x, 2];
RealDigits[sigma4, 10, 102][[1]]

polrootsreal(x^14 - x^12 - x^7 - x^2 + 1)[2] \\ Charles R Greathouse IV, Feb 11 2025

A316605 Decimal expansion of the fifth smallest known Salem number.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 08 2018

			1.21639166113826509162680631119946332772225360657057075756042706583831...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 10.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3), A306079 (sigma4), A316606 (sigma6), A316607 (sigma7), A316608 (sigma8), A316609 (sigma9), A316610 (sigma10).

c1 = {1, 0, 0, 0, -1, -1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2] - 1]).c2;
sigma5 = Root[p, x, 2];
RealDigits[sigma5, 10, 102][[1]]

polrootsreal(1 - x^4 - x^5 - x^6 + x^10)[2] \\ Charles R Greathouse IV, Feb 11 2025

p = 1 - x^4 - x^5 - x^6 + x^10.

A316606 Decimal expansion of the sixth smallest known Salem number.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 08 2018

			1.219720859040311844169606760414677944390415505541569678287974417873...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 18.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3 ), A306079 (sigma4), A316605 (sigma5), A316607 (sigma7), A316608 (sigma8), A316609 (sigma9), A316610 (sigma10).

c1 = {1, -1, 0, 0, 0, 0, 0, 0, -1, 1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2] - 1]).c2;
sigma6 = Root[p, x, 2];
RealDigits[sigma6, 10, 102][[1]]

polrootsreal(1 - x - x^8 + x^9 - x^10 - x^17 + x^18)[2] \\ Charles R Greathouse IV, Feb 11 2025

Equals root of p = 1 - x - x^8 + x^9 - x^10 - x^17 + x^18 with largest absolute value.

A316607 Decimal expansion of the seventh smallest known Salem number.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 08 2018

			1.2303914344072247027901779389752790175665744896617562414019142361728...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 10.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3 ), A306079 (sigma4), A316605 (sigma5), A316606 (sigma6), A316608 (sigma8), A316609 (sigma9), A316610 (sigma10).

c1 = {1, 0, 0, -1, 0, -1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2] - 1]).c2;
sigma7 = Root[p, x, 2];
RealDigits[sigma7, 10, 102][[1]]

polrootsreal(1 - x^3 - x^5 - x^7 + x^10)[2] \\ Charles R Greathouse IV, Feb 11 2025

p = 1 - x^3 - x^5 - x^7 + x^10.

cp's OEIS Frontend

A261406 Numbers n such that the norm of Phi_n(alpha), the n-th cyclotomic polynomial evaluated at alpha, is +-1, where alpha is the Salem number defined in A073011.

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A261407 Numbers d_m arising in Cohen et al.'s investigation of "ladder relations" among powers of the Salem number A073011.

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A029826 Expansion of 1/(x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1) (inverse of Salem polynomial).

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A181600 Expansion of 1/(1 - x - x^2 + x^8 - x^10).

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A219300 Decimal expansion of the second smallest known Salem number.

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A306078 Decimal expansion of the third smallest known Salem number.

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A306079 Decimal expansion of the fourth smallest known Salem number.

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A316605 Decimal expansion of the fifth smallest known Salem number.

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A316606 Decimal expansion of the sixth smallest known Salem number.