A374751 -id:A374751 - cp's OEIS frontend

A127583 Decimal expansion of smallest univoque Pisot Number.

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

Offset: 1

R. J. Mathar, Jun 13 2007

The 2nd and 3rd smallest univoque Pisot numbers are 1.8866814375493413002581625771460337570154853175085... (A374750) and 1.9051661677540189095727878303640157935069696492981... (A374751).

			1.88000047865555118281806896989947911064560839482260578509980692092...

J.-P. Allouche, C. Frougny, K. G. Hare, On univoque Pisot numbers, arXiv:math/0610681 [math.NT], (23-October-2006)
J.-P. Allouche, C. Frougny, K. G. Hare, On univoque Pisot numbers, Math. Comp. 76 (2007), 1639-1660.
D. Terr and Eric Weisstein's World of Mathematics, Pisot Numbers.
Index entries for algebraic numbers, degree 14.

Cf. A374750, A374751.

First@ RealDigits[ FindRoot[ x^14 - 2 x^13 + x^11 - x^10 - x^7 + x^6 - x^4 + x^3 - x + 1, {x, 1.88}, WorkingPrecision -> 111][[1, 2]]] (* Robert G. Wilson v, Jul 17 2007 *)
RealDigits[Root[x^14-2x^13+x^11-x^10-x^7+x^6-x^4+x^3-x+1,2],10,120][[1]] (* Harvey P. Dale, Jan 18 2024 *)

The largest real root of x^14 - 2*x^13 + x^11 - x^10 - x^7 + x^6 - x^4 + x^3 - x + 1. - Robert G. Wilson v, Jul 17 2007 (corrected, Joerg Arndt, Jun 13 2015)

More terms from Robert G. Wilson v, Jul 17 2007

A374750 Decimal expansion of the second smallest univoque Pisot number.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 18 2024

			1.8866814375493413002581625771460337570154853175...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Jean-Paul Allouche, Christiane Frougny, and Kevin G. Hare, On Univoque Pisot Numbers, Mathematics of Computation, Vol. 76, No. 259, July 2007, pp. 1639-1660 (arXiv version).
Eric Weisstein's World of Mathematics, Pisot Number.
Wikipedia, Pisot-Vijayaraghavan number.
Index entries for algebraic numbers, degree 12.

Cf. A127583 (smallest), A374751 (third smallest).

First[RealDigits[Root[#^12 - 2*#^11 + #^10 - 2*#^9 + #^8 - #^3 + #^2 - # + 1 &, 2], 10, 100]]

Equals the real root > 1 of x^12 - 2*x^11 + x^10 - 2*x^9 + x^8 - x^3 + x^2 - x + 1.

A374752 Decimal expansion of phi_4, a limit point of the set of Pisot numbers in (1,2).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 18 2024

			1.933184981899520446791424030335631586375183784479...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Jean-Paul Allouche, Christiane Frougny, and Kevin G. Hare, On Univoque Pisot Numbers, Mathematics of Computation, Vol. 76, No. 259, July 2007, pp. 1639-1660 (arXiv version).
Kevin G. Hare and Nikita Sidorov, Conjugates of Pisot numbers, International Journal of Number Theory, Vol. 17, No. 06 (2021), pp. 1307-1321 (arXiv version).
Eric Weisstein's World of Mathematics, Pisot Number.
Wikipedia, Pisot-Vijayaraghavan number.
Index entries for algebraic numbers, degree 5.

Cf. A001622, A058265, A086088, A109134, A275828, A374751.

First[RealDigits[Root[#^5 - 2*#^4 + # - 1 &, 1], 10, 100]]

Equals the real root of x^5 - 2*x^4 + x - 1.

Using the notation of Hare and Sidorov (2021, see Theorem 3.1), phi_1 = psi_1 (A001622) < phi_2 (A109134) < psi_2 (A058265) < phi_3 (A275828) < chi (A374751) < psi_3 (A086088) < phi_4 (this constant) < ... < psi_r < phi_(r+1) < ... < 2.

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A127583 Decimal expansion of smallest univoque Pisot Number.

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A374750 Decimal expansion of the second smallest univoque Pisot number.

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A374752 Decimal expansion of phi_4, a limit point of the set of Pisot numbers in (1,2).

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