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 1-4 of 4 results.

A083653 Consider the binary Champernowne sequence (A030190): smallest number m such that in binary representation n is contained in the concatenation of m and its successors.

Original entry on oeis.org

0, 1, 2, 1, 4, 2, 1, 3, 8, 4, 10, 2, 3, 1, 3, 7, 16, 8, 4, 9, 18, 10, 21, 2, 7, 3, 9, 1, 3, 5, 7, 15, 32, 16, 8, 17, 36, 4, 9, 19, 34, 18, 10, 10, 37, 21, 2, 6, 15, 7, 3, 12, 19, 9, 21, 1, 7, 3, 19, 5, 7, 13, 15, 31, 64, 32, 16, 33, 8, 34, 17, 35, 68, 36, 18, 4, 73, 9, 19, 39, 66, 34, 20, 18
Offset: 0

Views

Author

Reinhard Zumkeller, May 01 2003

Keywords

Comments

a(n)<=n; see A083655 for numbers m with a(m)=m;
a(A055143(n))=1;
A083654(n)-1 = number of successors of a(n) to cover n.

Examples

			n=24: '11000'=24 is a suffix of the concatenation of the first 8 numbers: '0'1'10'11'100'101'110'111'1000', therefore a(24)=7 and A083654(24)=2.

Links

Crossrefs

Cf. A030190, A030304, A007088, A345672 (decimal analog).
Cf. A055143, A083654, A083655.

Programs

  • PARI
    See Links section.

Extensions

Edited by Charles R Greathouse IV, Apr 26 2010

A083654 Consider the binary Champernowne sequence (A030190): number of successive numbers to be concatenated beginning with A083653(n) such that in binary representation n is contained.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 3, 2, 4, 2, 3, 2, 3, 2, 3, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 2, 2, 3, 2, 2, 2
Offset: 0

Views

Author

Reinhard Zumkeller, May 01 2003

Keywords

Comments

a(2^k)=1, see A083655 for all numbers m with a(m)=1;

Examples

			n=24: '11000'=24 is a suffix of the concatenation of the first 8 numbers: '0'1'10'11'100'101'110'111'1000', therefore a(24)=2 and A083653(24)=7.

Crossrefs

Cf. A030304, A007088.

A161374 "Punctual" binary numbers. Complement of A161373.

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 16, 22, 32, 36, 64, 128, 136, 256, 512, 528, 1024, 2048, 2080, 4096, 8192, 8256, 16384, 32768, 32896, 65536, 131072, 131328, 262144, 524288, 524800, 1048576, 2097152, 2098176, 4194304, 8388608, 8390656, 16777216, 33554432
Offset: 1

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Jun 08 2009

Keywords

Comments

A161373 U {a(n)} = A000027.
Whether or not 22 is punctual or early bird is a matter interpretation of "early occurrence" in the definition of A161373: 10110 occurs as the right 3 bits of 21 (10101) and the left 2 bits of 22 (10110) itself, which is ahead of the natural position, but not *completely* ahead of it. One can show (see weblink) the 22 is the only such case of doubt. [From Hagen von Eitzen, Jun 29 2009]

Links

Crossrefs

Cf. A083655, A116700, A161373.

Formula

From Hagen von Eitzen, Jun 29 2009: (Start)
G.f.: (1+x+2x^2)/(2-8x^3) + x/(2-4x^3) -1/2 -x + x^4 + 4x^5 + 2x^6 + 6x^7 + 6x^8
If q>=3 then a(3q) = 2^(2q-1), a(3q+1) = 2^(2q-1) + 2^(q-1), a(3q+2) = 2^(2q). (End)
a(n) = A083655(n-2) for n>=9. - Alois P. Heinz, Dec 14 2022

Extensions

Offset corrected as customary for lists, 20 removed by Hagen von Eitzen, Jun 27 2009
More terms from Hagen von Eitzen, Jun 29 2009

A296365 Numbers which appear prematurely in the binary Champernowne word (A030190).

Original entry on oeis.org

3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85
Offset: 1

Views

Author

N. J. A. Sloane, Dec 17 2017

Keywords

Crossrefs

This is the complement of A083655.
Cf. A030190, A161373.
Showing 1-4 of 4 results.

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

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