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.

Previous Showing 31-38 of 38 results.

A378330 Decimal expansion of the base 6 Champernowne constant.

Original entry on oeis.org

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

Views

Author

Joshua Searle, Nov 23 2024

Keywords

Comments

This constant is formed by the concatenation of the natural numbers in base 6 and then converted into base 10.
This constant is 6-normal.

Examples

			0.239862685815066767447719828672209624590576971529350213760693195631576583...

Links

Crossrefs

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

Programs

  • Mathematica
    First[RealDigits[ChampernowneNumber[6], 10, 100]]

A378331 Decimal expansion of the base 7 Champernowne constant.

Original entry on oeis.org

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

Views

Author

Joshua Searle, Nov 25 2024

Keywords

Comments

This constant is formed by the concatenation of the natural numbers in base 7 and then converted into base 10.
This constant is 7-normal.

Examples

			0.194435535086240521475840093082908576452932971050422112479588531233679088...

Links

Crossrefs

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307
(base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

Programs

  • Mathematica
    First[RealDigits[ChampernowneNumber[7], 10, 100]]

A378332 Decimal expansion of the base 8 Champernowne constant.

Original entry on oeis.org

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

Views

Author

Joshua Searle, Nov 25 2024

Keywords

Comments

This constant is formed by the concatenation of the natural numbers in base 8 and then converted into base 10.
This constant is 8-normal.

Examples

			0.163264812105216797367094986142605190224237843285462333081380700428319475...

Links

Crossrefs

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

Programs

  • Mathematica
    First[RealDigits[ChampernowneNumber[8], 10, 100]]

A378333 Decimal expansion of the base 9 Champernowne constant.

Original entry on oeis.org

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

Views

Author

Joshua Searle, Nov 25 2024

Keywords

Comments

This constant is formed by the concatenation of the natural numbers in base 9 and then converted into base 10.
This constant is 9-normal.

Examples

			0.140624976119696782479669008935663183265457083246828486657555171275414914...

Links

Crossrefs

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

Programs

  • Mathematica
    First[RealDigits[ChampernowneNumber[9], 10, 100]]

A332412 a(n) is the real part of f(n) = Sum_{d_k > 0} 3^k * i^(d_k-1) where Sum_{k >= 0} 5^k * d_k is the base 5 representation of n and i denotes the imaginary unit. Sequence A332413 gives imaginary parts.

Original entry on oeis.org

0, 1, 0, -1, 0, 3, 4, 3, 2, 3, 0, 1, 0, -1, 0, -3, -2, -3, -4, -3, 0, 1, 0, -1, 0, 9, 10, 9, 8, 9, 12, 13, 12, 11, 12, 9, 10, 9, 8, 9, 6, 7, 6, 5, 6, 9, 10, 9, 8, 9, 0, 1, 0, -1, 0, 3, 4, 3, 2, 3, 0, 1, 0, -1, 0, -3, -2, -3, -4, -3, 0, 1, 0, -1, 0, -9, -8, -9
Offset: 0

Views

Author

Rémy Sigrist, Feb 12 2020

Keywords

Comments

The representation of {f(n)} corresponds to the cross form of the Vicsek fractal.
As a set, {f(n)} corresponds to the Gaussian integers whose real and imaginary parts have not simultaneously a nonzero digit at the same place in their balanced ternary representations.

Examples

			For n = 103:
- 103 = 4*5^2 + 3*5^0,
- so f(123) = 3^2 * i^(4-1) + 3^0 * i^(3-1) = -1 - 9*i,
- and a(n) = -1.

Links

Crossrefs

See A332497 for a similar sequence.
Cf. A031219, A289813, A332413 (imaginary parts).

Programs

  • PARI
    a(n) = { my (d=Vecrev(digits(n,5))); real(sum (k=1, #d, if (d[k], 3^(k-1)*I^(d[k]-1), 0))) }

Formula

a(n) = 0 iff the n-th row of A031219 has only even terms.
a(5*n) = 3*a(n).
a(5*n+1) = 3*a(n) + 1.
a(5*n+2) = 3*a(n).
a(5*n+3) = 3*a(n) - 1.
a(5*n+4) = 3*a(n).

A332413 a(n) is the imaginary part of f(n) = Sum_{d_k > 0} 3^k * i^(d_k-1) where Sum_{k >= 0} 5^k * d_k is the base 5 representation of n and i denotes the imaginary unit. Sequence A332412 gives real parts.

Original entry on oeis.org

0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 3, 3, 4, 3, 2, 0, 0, 1, 0, -1, -3, -3, -2, -3, -4, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 3, 3, 4, 3, 2, 0, 0, 1, 0, -1, -3, -3, -2, -3, -4, 9, 9, 10, 9, 8, 9, 9, 10, 9, 8, 12, 12, 13, 12, 11, 9, 9, 10, 9, 8, 6, 6, 7, 6, 5, 0, 0, 1, 0
Offset: 0

Views

Author

Rémy Sigrist, Feb 12 2020

Keywords

Examples

			For n = 103:
- 103 = 4*5^2 + 3*5^0,
- so f(123) = 3^2 * i^(4-1) + 3^0 * i^(3-1) = -1 - 9*i,
- and a(n) = -9.

Links

Crossrefs

Cf. A031219, A289814, A332412 (real parts and additional comments).

Programs

  • PARI
    a(n) = { my (d=Vecrev(digits(n,5))); imag(sum (k=1, #d, if (d[k], 3^(k-1)*I^(d[k]-1), 0))) }

Formula

a(n) = 0 iff the n-th row of A031219 has neither 2's nor 4's.
a(5*n) = 3*a(n).
a(5*n+1) = 3*a(n).
a(5*n+2) = 3*a(n) + 1.
a(5*n+3) = 3*a(n).
a(5*n+4) = 3*a(n) - 1.

A275993 Champernowne sequence: write n in base 16 and juxtapose.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 1, 14, 1, 15, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 2, 11, 2, 12, 2, 13, 2, 14, 2, 15, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 3, 11, 3
Offset: 0

Views

Author

Robert G. Wilson v, Aug 15 2016

Keywords

Comments

10 -> A, 11 -> B, 12 -> C, 13 -> D, 14 -> E & 15 -> F.

Links

Crossrefs

Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), A003137 and A054635 (b=3), A030373 (b=4), A031219 (b=5), A030548 (b=6), A030998 (b=7), A031035 and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10).

Programs

  • Mathematica
    almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b -1) i*b^(i -1) + l; i++]; i--; p = Mod[d -l, i]; q = Floor[(d -l)/i] + b^(i -1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q -1, b]]]; Array[ almostNatural[#, 16] &, 105, 0]
First[RealDigits[ChampernowneNumber[16], 16, 100, 0]] (* Paolo Xausa, Jun 21 2024 *)

A031242 Length of n-th run of digit 0 in A031235.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Differs from A031226 whenever a base 5 sequence of a number is inserted which has a non-palindromic sequence of run-lengths of 0. Happens first scanning 10010_5 = 630_10 which is inserted in A031235 as 01001 (run lengths 1,2) but in A031219 as 10010 (run lengths 2,1). - R. J. Mathar, Jul 23 2025

Extensions

a(1)=1 inserted for consistency with change in A031235 by Sean A. Irvine, Apr 19 2020
Previous Showing 31-38 of 38 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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