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 1-7 of 7 results.

A255059 Positions n where A255054(n) is odd.

Original entry on oeis.org

0, 2, 3, 5, 6, 10, 11, 14, 15, 19, 20, 23, 24, 25, 26, 30, 31, 35, 36, 39, 40, 41, 42, 44, 48, 52, 53, 54, 55, 59, 60, 64, 65, 68, 69, 70, 71, 73, 77, 79, 81, 83, 88, 92, 93, 94, 95, 97, 101, 105, 106, 107, 108, 112, 113, 117, 118, 121, 122, 123, 124, 126, 130, 132, 134, 136, 141, 143, 145, 148, 149, 151, 154, 156, 161, 165
Offset: 1

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Author

Antti Karttunen, Feb 16 2015

Keywords

Links

Crossrefs

Complement: A255060.
Cf. A255061 (apart from its second term 1 is a subsequence).
Cf. A255054.

A255060 Positions n where A255054(n) is even.

Original entry on oeis.org

1, 4, 7, 8, 9, 12, 13, 16, 17, 18, 21, 22, 27, 28, 29, 32, 33, 34, 37, 38, 43, 45, 46, 47, 49, 50, 51, 56, 57, 58, 61, 62, 63, 66, 67, 72, 74, 75, 76, 78, 80, 82, 84, 85, 86, 87, 89, 90, 91, 96, 98, 99, 100, 102, 103, 104, 109, 110, 111, 114, 115, 116, 119, 120, 125, 127, 128, 129, 131, 133, 135, 137, 138, 139, 140, 142, 144
Offset: 1

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Author

Antti Karttunen, Feb 16 2015

Keywords

Links

Crossrefs

Complement: A255059.
Cf. A255054.

A255061 Number of steps to reach 0 when starting from (2^n)-2 and iterating the map x -> x - (number of runs in binary representation of x): a(n) = A255072(A000918(n)).

Original entry on oeis.org

0, 1, 3, 6, 11, 20, 36, 65, 118, 215, 393, 721, 1329, 2463, 4589, 8590, 16142, 30434, 57549, 109114, 207388, 395045, 754027, 1441971, 2762764, 5303466, 10200385, 19656528, 37948281, 73384080, 142115376, 275551755, 534790472, 1038702980, 2018626772, 3924923937, 7634733312
Offset: 1

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Author

Antti Karttunen, Feb 14 2015

Keywords

Comments

Apart from a(1)=1, also gives the positions of ones in A255054.

Crossrefs

One less than A255062.
First differences: A255071.
Apart from a(1)=1, a subsequence of A255059.
Cf. A000918, A255072, A255054.
Analogous sequences: A218600, A226061.

Programs

Formula

a(n) = A255072(A000918(n)).
a(1) = 0; for n > 1, a(n) = a(n-1) + A255071(n-1).
Other identities. For all n >= 1:
a(n) = A255062(n) - 1.

A255072 Number of steps to reach 0 starting with n and using the iterated process: x -> x - (number of runs in binary representation of x).

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 5, 5, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 25
Offset: 0

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Author

Antti Karttunen, Feb 14 2015

Keywords

Links

Crossrefs

Cf. A255053 (least inverse), A255055 (greatest inverse), A255054 (run lengths).
Cf. A005811, A236840, A255071, A255056.
Cf. A255061 & A255062 (values at points (2^n)-2 and (2^n)-1).
Analogous sequences: A071542, A219642, A219652

Formula

a(0) = 0; for n >= 1, a(n) = 1 + a(A236840(n)) = 1 + a(n - A005811(n)).
Other identities. For all n >= 0:
a(A255053(n)) = a(A255055(n)) = n.
a(A255056(n)) = n. [This sequence works also as an inverse function for number-of-runs beanstalk A255056.]

A255070 (1/2)*(n minus number of runs in the binary expansion of n): a(n) = (n - A005811(n)) / 2 = A236840(n)/2.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 3, 3, 3, 3, 4, 5, 5, 6, 7, 7, 7, 7, 8, 8, 8, 9, 10, 11, 11, 11, 12, 13, 13, 14, 15, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 18, 19, 20, 20, 21, 22, 23, 23, 23, 24, 24, 24, 25, 26, 27, 27, 27, 28, 29, 29, 30, 31, 31, 31, 31, 32, 32, 32, 33, 34, 34, 34, 34, 35
Offset: 0

Views

Author

Antti Karttunen, Feb 14 2015

Keywords

Links

Crossrefs

Least inverse: A091067 (also the positions of records).
Greatest inverse: A255068.
Run lengths: A106836.
Cf. A005811, A060833, A236840, A255054, A255056, A255072, A269363, A269367.

Programs

  • Mathematica
    a[n_] := (n - Length@ Split[IntegerDigits[n, 2]])/2; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Jul 16 2023 *)
  • Scheme
    (define (A255070 n) (/ (A236840 n) 2))

Formula

a(n) = A236840(n) / 2 = (n - A005811(n)) / 2.
Other identities:
a(A091067(n)) = n for all n >= 1.
a(A255068(n)) = n for all n >= 0.
a(A269363(n)) = A269367(n). - Antti Karttunen, Aug 12 2019

A255053 Least inverse of A255072; a(n) = smallest k such that A255072(k) = n.

Original entry on oeis.org

0, 1, 3, 6, 7, 11, 14, 15, 19, 23, 27, 30, 31, 35, 39, 44, 47, 51, 55, 59, 62, 63, 67, 71, 76, 79, 86, 91, 95, 99, 103, 108, 111, 115, 119, 123, 126, 127, 131, 135, 140, 143, 150, 155, 159, 166, 172, 176, 182, 187, 191, 195, 199, 204, 207, 214, 219, 223, 227, 231, 236, 239, 243, 247, 251, 254, 255
Offset: 0

Views

Author

Antti Karttunen, Feb 14 2015

Keywords

Comments

Also positions of records in A255072.

Links

Crossrefs

Cf. A255072, A255054, A255055, A255056, A255123.
Analogous sequences: A213708, A219643, A219653.

Formula

Other identities. For all n >= 0:
a(0) = 0; for n > 0: a(n) = a(n-1) + A255054(n-1).
a(n) = A255056(n) - A255123(n).

A255055 Greatest inverse of A255072; a(n) = largest k such that A255072(k) = n.

Original entry on oeis.org

0, 2, 5, 6, 10, 13, 14, 18, 22, 26, 29, 30, 34, 38, 43, 46, 50, 54, 58, 61, 62, 66, 70, 75, 78, 85, 90, 94, 98, 102, 107, 110, 114, 118, 122, 125, 126, 130, 134, 139, 142, 149, 154, 158, 165, 171, 175, 181, 186, 190, 194, 198, 203, 206, 213, 218, 222, 226, 230, 235, 238, 242, 246, 250, 253, 254, 258
Offset: 0

Views

Author

Antti Karttunen, Feb 14 2015

Keywords

Links

Crossrefs

Cf. A255053, A255054, A255072.
Analogous sequences: A173601, A219645, A219655.

Formula

a(0) = 0; for n > 0, a(n) = A255054(n) + a(n-1).
Other identities. For all n >= 0:
a(n) = A255053(n) + A255054(n) - 1.
a(n) = A255056(n) + A255124(n).
Showing 1-7 of 7 results.

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