A059991 -id:A059991 - cp's OEIS frontend

A060481 Number of orbits of length n in a map whose periodic points come from A059991.

1, 0, 1, 0, 3, 2, 9, 0, 28, 24, 93, 20, 315, 288, 1091, 0, 3855, 3626, 13797, 3264, 49929, 47616, 182361, 2720, 671088, 645120, 2485504, 599040, 9256395, 8947294, 34636833, 0, 130150493, 126320640, 490853403, 119302820, 1857283155, 1808400384, 7048151355

Offset: 1

Thomas Ward

From Petros Hadjicostas, Jan 15 2018: (Start)
Terms a(2)-a(20) of this sequence and sequence A000048 appear on p. 311 of Sommerville (1909) in the context of sequences of "evolutes" of cyclic compositions of positive integers.
Algebraically, it is easy to prove that this sequence and sequence A000048 have the same odd-indexed terms. (End)

V. Chothi, G. Everest, and T. Ward, S-integer dynamical systems: periodic points, J. Reine Angew. Math., 489 (1997), 99-132.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
D. M. Y. Sommerville, On certain periodic properties of cyclic compositions of numbers, Proc. London Math. Soc. S2-7(1) (1909), 263-313.
T. Ward, Almost all S-integer dynamical systems have many periodic points, Ergodic Theory and Dynamical Systems, 18 (1998), 471-486.

Cf. A000048, A059991.

If b(n) is the n-th term of A059991, then a(n) = (1/n)* Sum_{d|n} mu(d)*b(n/d). [Corrected by Petros Hadjicostas, Jan 14 2018]
From Petros Hadjicostas, Jan 14 2018: (Start)
a(2*n-1) = A000048(2*n-1) for n >= 1.
a(2^m) = 0 for m >= 1.
G.f.: If B(x) is the g.f. of the sequence b(n) = A059991(n) and C(x) = integrate(B(y)/y, y = 0..x), then the g.f. of the current sequence is A(x) = Sum_{n>=1} (mu(n)/n)*C(x^n). (End)

a(18)-a(30) by Petros Hadjicostas, Jan 15 2018

A129760 Bitwise AND of binary representation of n-1 and n.

Original entry on oeis.org

0, 0, 2, 0, 4, 4, 6, 0, 8, 8, 10, 8, 12, 12, 14, 0, 16, 16, 18, 16, 20, 20, 22, 16, 24, 24, 26, 24, 28, 28, 30, 0, 32, 32, 34, 32, 36, 36, 38, 32, 40, 40, 42, 40, 44, 44, 46, 32, 48, 48, 50, 48, 52, 52, 54, 48, 56, 56, 58, 56, 60, 60, 62, 0, 64, 64, 66, 64, 68, 68, 70, 64, 72, 72, 74

Offset: 1

Russ Cox, May 15 2007

Also the number of Ducci sequences with period n.
Also largest number less than n having in binary representation fewer ones than n has; A048881(n-1) = A000120(a(n)) = A000120(n)-1. - Reinhard Zumkeller, Jun 30 2010
a(n) is the parent of vertex n in the binomial tree.  The binomial tree is root vertex n=0, then for n>=1 the parent of n is n with its least significant 1-bit changed to a 0-bit.  Binomial tree order 5, n=0 to 31 inclusive, is the frontispiece of Knuth volume 1, second and subsequent editions.  Vertices are shown there with n in binary dots and a(n) is the next vertex towards the root at the bottom of the page. - Kevin Ryde, Jul 24 2019

			a(6) = 6 AND 5 = binary 110 AND 101 = binary 100 = 4.

Donald E. Knuth, The Art of Computer Programming, volume 1, second edition, frontispiece. Reproduced with brief description of the art in Donald E. Knuth, Selected Papers on Fun and Games, 2010, Chapter 47 Geek Art, figure 16, page 679.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ron Brown and Jonathan L. Merzel, The number of Ducci sequences with a given period, Fib. Quart., 45 (2007), 115-121.

Cf. A038712, A086799, A104594, A059991, A006519, A109168, A220466.

C
```
int a(int n) { return n & (n-1); }
```

[n - 2^Valuation(n, 2): n in [1..100]]; // Vincenzo Librandi, Jul 25 2019

nmax := 75: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (2*n-2) * 2^p od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jun 22 2011, revised Jan 25 2013
A129760 := n -> Bits:-And(n-1, n):
seq(A129760(n), n=1..75); # Peter Luschny, Sep 26 2019

Table[BitAnd[n, n - 1], {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)

a(n)=bitand(n,n-1) \\ Charles R Greathouse IV, Jun 23 2011

def a(n): return n & (n-1)
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Jul 13 2022

a(n) = n AND n-1.
Equals n - A006519(n). - N. J. A. Sloane, May 26 2008
From Johannes W. Meijer, Jun 22 2011: (Start)
a((2*n-1)*2^p) = (2*n-2)*(2^p), p>=0.
a(2*n-1) = (2*n-2), n>=1, and a(2^p+1) = 2^p, p>=1. (End)

A059990 Number of points of period n under the dual of the map x->2x on Z[1/6].

Original entry on oeis.org

1, 1, 7, 5, 31, 7, 127, 85, 511, 341, 2047, 455, 8191, 5461, 32767, 21845, 131071, 9709, 524287, 349525, 2097151, 1398101, 8388607, 1864135, 33554431, 22369621, 134217727, 89478485, 536870911, 119304647

Offset: 1

Thomas Ward, Mar 08 2001

This sequence counts the periodic points in the simplest nontrivial S-integer dynamical system. These dynamical systems arise naturally in arithmetic and are built by making an isometric extension of a familiar hyperbolic system. The extension destroys some of the periodic points, in this case reducing the original number 2^n-1 by factoring out any 3's. An interesting feature is that the logarithmic growth rate is still log 2.

A059990[n+7] times some power of 3 seems to me the greatest common Denominator of A035522[4n+16+1],A035522[4n+16+2],A035522[4n+16+3] and A035522[4n+16+4] for n>1 [From Dylan Hamilton, Aug 04 2010]

			a(6)=7 because 2^6-1 = 3^2x7, so |2^6-1|_3=3^(-2).

V. Chothi, G. Everest, T. Ward. S-integer dynamical systems: periodic points. J. Reine Angew. Math., 489 (1997), 99-132.
T. Ward. Almost all S-integer dynamical systems have many periodic points. Erg. Th. Dynam. Sys. 18 (1998), 471-486.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Cf. A000225, A001945, A059991.

a(n)=(2^n-1)x|2^n-1|_3

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A060481 Number of orbits of length n in a map whose periodic points come from A059991.

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A129760 Bitwise AND of binary representation of n-1 and n.

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A059990 Number of points of period n under the dual of the map x->2x on Z[1/6].

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