A131136 -id:A131136 - cp's OEIS frontend

A063787 a(2^k) = k + 1 and a(2^k + i) = 1 + a(i) for k >= 0 and 0 < i < 2^k.

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 3, 4, 4, 5, 4, 5, 5, 6, 4

Offset: 1

Reinhard Zumkeller, Aug 16 2001

nonn

Hamming weights of odd numbers. - Friedjof Tellkamp, Jan 11 2024

			k = 3: a(2^3) = a(8) = 4 = 3 + 1.
k = 3, i = 5: a(2^3 + 5) = a(13) = 3 = 1 + 2 = 1 + a(5).
From _Omar E. Pol_, Jun 12 2009: (Start)
Triangle begins:
  1;
  2,2;
  3,2,3,3;
  4,2,3,3,4,3,4,4;
  5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5;
  6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6;
  7,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,...
(End)

Michael Gilleland, Some Self-Similar Integer Sequences

Cf. A000079, A000120, A007814, A086799, A047457, A131136.

Cf. A330038 (partial sums).

Table[DigitCount[2 n - 1, 2, 1], {n, 1, 105}] (* Friedjof Tellkamp, Jan 11 2024 *)

a(n) = hammingweight(n-1) + 1; \\ Michel Marcus, Nov 23 2022

def a(n): return bin(n-1).count('1') + 1
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Dec 16 2021

a(n) = A000120(n-1) + 1.
a(n) = log(A131136)/log(2). - Stephen Crowley, Aug 25 2008
a(n) = A007814(n) + A000120(n). - Gary W. Adamson, Jun 04 2009
a(n) = A000120(A086799(n)). - Reinhard Zumkeller, Jul 31 2010
a(n) = A000120(A047457(n)-1) = A000120(A047457(n)+1). - Ilya Lopatin, Mar 16 2014
a(n) = A000120(2n-1). - Friedjof Tellkamp, Jan 11 2024

A080978 a(n) = 2*A006046(n) + 1.

Original entry on oeis.org

1, 3, 7, 11, 19, 23, 31, 39, 55, 59, 67, 75, 91, 99, 115, 131, 163, 167, 175, 183, 199, 207, 223, 239, 271, 279, 295, 311, 343, 359, 391, 423, 487, 491, 499, 507, 523, 531, 547, 563, 595, 603, 619, 635, 667, 683, 715, 747, 811, 819, 835, 851, 883, 899, 931, 963

Offset: 0

Antti Karttunen, Mar 02 2003

nonn

The number of edges in A080973-trees.

Conjectured partial sums of A131136. - Sean A. Irvine, Jun 25 2022

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 6, 30.

Cf. A006046, A080973.

Map[2 # + 1 &, {0}~Join~Accumulate@ Map[Count[#, ?OddQ] &, Table[Binomial[n, k], {n, 0, 54}, {k, 0, n}]]] (* _Michael De Vlieger, Oct 30 2022 *)

A131137 Denominator of (exponential) expansion of log((2*x/3-1)/(x-1)).

Original entry on oeis.org

1, 3, 9, 27, 27, 81, 243, 243, 729, 2187, 729, 2187, 6561, 6561, 19683, 59049, 59049, 177147, 531441, 177147, 531441, 1594323, 1594323, 4782969, 14348907, 14348907, 43046721, 129140163, 14348907, 43046721, 129140163, 129140163, 387420489

Offset: 0

Paul Barry, Jun 17 2007

a(n) = 3^A131138(n).
Also, starting at second term, denominator of (1-(2/3)^n)*(n-1)!;
Conjecture: starting at third term, also equals the denominator of polylog(-n,1/4)/4. - Wouter Meeussen, Feb 13 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A131136.

Denominator[CoefficientList[Series[Log[(2 x/3 - 1)/(x - 1)], {x, 0, 32}], x] Range[0, 32]!]; (* or *) Prepend[Table[Denominator[(1 - (2/3)^n) (n - 1)!], {n, 32}], 1]; (* or *) Join[{1, 3}, Table[Denominator[PolyLog[-n, 1/4]/4 ], {n, 31}]] (* Wouter Meeussen, Feb 13 2014 *)

a(n)=if(n<4,3^n,denominator(polylog(1-n,1/4)/4)) \\ Charles R Greathouse IV, Jul 15 2014

A131135 Denominator of (ordinary) expansion of log((x/2-1)/(x-1)).

Original entry on oeis.org

1, 2, 8, 24, 64, 160, 128, 896, 2048, 4608, 10240, 22528, 16384, 106496, 229376, 491520, 1048576, 2228224, 524288, 9961472, 4194304, 6291456, 92274688, 192937984, 134217728, 838860800, 1744830464, 3623878656, 7516192768

Offset: 0

Paul Barry, Jun 17 2007

Cf. A090634, A131136.

a(n) = 2*A090634(n) for n > 0.

a(n) = denominator((1-1/2^n)/n), for n > 0, conjectured. - Michel Marcus, Sep 12 2019

cp's OEIS Frontend

A063787 a(2^k) = k + 1 and a(2^k + i) = 1 + a(i) for k >= 0 and 0 < i < 2^k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A080978 a(n) = 2*A006046(n) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A131137 Denominator of (exponential) expansion of log((2*x/3-1)/(x-1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A131135 Denominator of (ordinary) expansion of log((x/2-1)/(x-1)).

Views

Author

Keywords

Crossrefs

Formula