cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A329337 Continued fraction of A328907 = 0.6009668516..., solution to 1 + 3^x = 6^x.

Original entry on oeis.org

0, 1, 1, 1, 1, 40, 1, 3, 2, 1, 2, 23, 1, 13, 1, 8, 1, 15, 2, 3, 103, 4, 10, 4, 2, 2, 2, 1, 1, 84, 1, 4, 1, 3, 1, 1, 5, 1, 7, 23, 8, 1, 8, 24, 1, 1, 2, 12, 39, 14, 19, 3, 4, 8, 3, 2, 1, 4, 1, 8, 1, 1, 1, 2, 1, 10, 1, 35, 1, 10, 2, 2, 2, 1, 1, 15, 2, 3, 1, 4, 7, 5, 1, 9, 1, 1, 1, 1, 2, 3, 3, 2, 1, 4, 54, 4, 1, 3, 2, 1, 1, 1, 1, 4, 22, 1, 4, 3, 1, 1, 1, 2, 6, 1, 1, 4, 1, 8, 1, 20
Offset: 0

Views

Author

M. F. Hasler, Nov 11 2019

Keywords

Examples

			0.6009668516... = 0 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(40 + 1/(1 + 1/(3 + ...)))))))

Crossrefs

Cf. A328912 (cont. frac. of A242208: 1 + 2^x = 4^x), A328913 (cont. frac. of A328900: 2^x + 3^x = 4^x), A329335 (cont. frac. of A328905: 1 + 2^x = 5^x).

Programs

  • Mathematica
    ContinuedFraction[x/.FindRoot[1+3^x==6^x,{x,1},WorkingPrecision->150]] (* Harvey P. Dale, Jun 13 2022 *)
  • PARI
    contfrac(c=solve(x=0,1, 1+3^x-6^x))[^-1] \\ discarding possibly incorrect last term. Use e.g. \p999 to get more terms. - M. F. Hasler, Oct 31 2019

A328900 Decimal expansion of s = 1.507126591638653..., solution to 2^s + 3^s = 4^s.

Original entry on oeis.org

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

Views

Author

M. F. Hasler, on suggestion from Artur Jasinski, Oct 30 2019

Keywords

Comments

Equivalently, solution to 1/(1 - 2^-s) = 1 + 2^-s + 3^-s, related to partial sums and Euler product approximating zeta(s).
When a + b = c, then the only solution to a^x + b^x = c^x is trivially x = 1. The solution to 1 + 2^x = 4^x is log_2(Phi) = A242208.
See A328904, A328905 for (a, b, c) = (1, 3, 5) and (1, 2, 5).

Examples

			1.5071265916386531339868833608386311643739940944856568966753643594438147338...

Links

Crossrefs

Cf. A328913 (continued fraction).
Cf. A242208 (1 + 2^x = 4^x), A328912 (continued fraction thereof).
Cf. A328904 (1 + 3^x = 5^x), A328905 (1 + 2^x = 5^x), A328906 (1 + 2^x = 6^x), A328907 (1 + 3^x = 6^x).

Programs

  • Mathematica
    RealDigits[ x /. FindRoot[2^x + 3^x == 4^x, {x, 1.5}, WorkingPrecision -> 100]][[1]] (* Artur Jasinski, Oct 30 2019 *)
  • PARI
    solve(s=1,2,2^s+3^s-4^s) \\ use e.g. \p200 to get more digits

A328906 Decimal expansion of the solution x = 0.4895363211996... to 1 + 2^x = 6^x.

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Nov 11 2019

Keywords

Examples

			0.489536321199649488689875316822650189403586515771912127846436678619255628...

Crossrefs

Cf. A329336 (continued fraction).
Cf. A242208 (1 + 2^x = 4^x), A328900 (2^x + 3^x = 4^x), A328904 (1 + 3^x = 5^x), A328905 (1 + 2^x = 5^x), A328907 (1 + 3^x = 6^x).

Programs

  • Mathematica
    RealDigits[x /. FindRoot[1 + 2^x == 6^x, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Jun 28 2023 *)
  • PARI
    print(c=solve(x=0,1, 1+2^x-6^x)); digits(c\.1^default(realprecision))[^-1] \\ [^-1] to discard possibly incorrect last digit. Use e.g. \p999 to get more digits. - M. F. Hasler, Oct 31 2019

A329334 Continued fraction of A328904 = 0.7271601514124259..., solution to 1 + 3^x = 5^x.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 72, 1, 3, 2, 6, 1, 1, 2, 45, 1, 5, 13, 73, 1, 2, 1, 9, 1, 1, 1, 3, 2, 3, 3, 2, 1, 2, 2, 1, 1, 19, 1, 1, 1, 1, 5, 1, 5, 2, 4, 3, 1, 6, 1, 1, 2, 1, 9, 8, 1, 4, 1, 1, 20, 1, 1, 2, 1, 5, 2, 2, 1, 2, 5, 1, 56, 1, 1, 1, 6, 127, 1, 1, 7, 2, 7, 1, 6, 1, 1, 3, 1, 54, 1, 1, 3, 2, 1, 1, 3
Offset: 0

Views

Author

M. F. Hasler, Nov 11 2019

Keywords

Examples

			0.7271601514124259... = 0 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + 1/(1  + 1/(72 + 1/...))))))

Crossrefs

Cf. A328912 (cont. frac. of A242208: 1 + 2^x = 4^x), A328913 (cont. frac. of A328900: 2^x + 3^x = 4^x), A329337 (cont. frac. of A328907: 1 + 3^x = 6^x).

Programs

  • PARI
    contfrac(c=solve(x=0,1, 1+3^x-5^x))[^-1] \\ discarding possibly incorrect last term. Use e.g. \p999 to get more terms. - M. F. Hasler, Oct 31 2019

A329335 Continued fraction of A328905 = 0.5638955242599..., solution to 1 + 2^x = 5^x.

Original entry on oeis.org

0, 1, 1, 3, 2, 2, 2, 1, 3, 3, 1, 5, 3, 1, 1, 3, 1, 1, 6, 1, 36, 20, 18, 3, 1, 3, 2, 1, 2, 9, 3, 2, 1, 1, 1, 2, 7, 1, 1, 5, 1, 112, 2, 1, 6, 2, 1, 1, 1, 1, 2, 44, 1, 2, 3, 70, 1, 1, 1, 12, 3, 1, 5, 6, 1, 1, 10, 4, 4, 2, 3, 1, 7, 1, 4, 1, 1, 1, 5, 2, 1, 5, 1, 4, 3, 1, 1, 1, 1, 2, 1, 1, 4, 6, 7, 2
Offset: 0

Views

Author

M. F. Hasler, Nov 11 2019

Keywords

Examples

			0.5638955242599... = 0 + 1/(1 + 1/(1 + 1/(3 + 1/(2 + 1/(2  + 1/(2 + 1/...))))))

Crossrefs

Cf. A328912 (cont. frac. of A242208: 1 + 2^x = 4^x), A328913 (cont. frac. of A328900: 2^x + 3^x = 4^x), A329337 (cont. frac. of A328907: 1 + 3^x = 6^x).

Programs

  • PARI
    contfrac(c=solve(x=0,1, 1+2^x-5^x))[^-1] \\ discarding possibly incorrect last term. Use e.g. \p999 to get more terms. - M. F. Hasler, Oct 31 2019
Showing 1-5 of 5 results.

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