A038553 -id:A038553 - cp's OEIS frontend

A111944 Number of different lengths of cycles for vectors of length n under the Ducci map.

1, 1, 2, 1, 2, 3, 2, 1, 3, 3, 2, 4, 2, 3, 4, 1, 3, 5, 2, 4, 5, 3, 2, 5, 3, 3, 4, 4, 2, 7, 2, 1, 4, 5, 6, 7, 2, 3, 6, 5, 3, 9, 2, 4, 9, 3, 2, 6, 3, 5, 4, 4, 2, 7, 6, 5, 4, 3, 2, 10, 2, 3, 6, 1, 5, 7, 2, 7, 6, 11, 2, 9, 3, 3, 7, 4, 6, 11, 2, 6, 5, 5, 2, 13, 6, 3, 5, 5, 2, 17, 5, 4, 6, 3, 6, 7, 3, 5, 7, 7

Offset: 1

N. J. A. Sloane, Nov 27 2005

nonn

Total number of cycles is given by A135547.

Max Alekseyev, Table of n, a(n) for n = 1..420
N. J. Calkin, J. G. Stevens, D. M. Thomas, A characterization for the lengths of cycles of the n-number Ducci game, Fib. Q. 43(1), 2005, 53-59.

Cf. A038553.

Terms a(1), a(2), and a(41) onward from Max Alekseyev, Oct 12 2013

A136043 Period-lengths of the base-2 MR-expansions of the reciprocals of the positive integers.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 5, 1, 6, 1, 1, 1, 4, 3, 9, 2, 2, 5, 4, 1, 10, 6, 9, 1, 14, 1, 1, 1, 5, 4, 5, 3, 18, 9, 4, 2, 10, 2, 7, 5, 5, 4, 9, 1, 10, 10, 2, 6, 26, 9, 8, 1, 9, 14, 29, 1, 30, 1, 1, 1, 6, 5, 33, 4, 11, 5, 14, 3, 3, 18, 9, 9, 15, 4, 17, 2, 27, 10, 41, 2, 2, 7, 11, 5, 4, 5, 4, 4, 3, 9, 14

Offset: 1

John W. Layman, Dec 12 2007

nonn

It appears that if p is a prime with 2 as a primitive root (A001122), then a(p)=(p-1)/2. This has been confirmed for primes up to 2000. See A136042 for the definition of the MR-expansion of a positive real number.

			In A136042 it is shown that the base-2 MR-expansion of 1/5 is {3,1,3,1,3,1,3,1,...}, with period-length 2, so a(5)=2.

Cf. A001122, A136042, A135044, A119513, A038553.

a(n) = 1 if n is a power of 2. It seems that otherwise a(n) = A119513(n). - Andrei Zabolotskii, Jul 28 2025

A119513 a(n) = A119957(n) / n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 0, 3, 2, 5, 1, 6, 1, 1, 0, 4, 3, 9, 2, 2, 5, 4, 1, 10, 6, 9, 1, 14, 1, 1, 0, 5, 4, 5, 3, 18, 9, 4, 2, 10, 2, 7, 5, 5, 4, 9, 1, 10, 10, 2, 6, 26, 9, 8, 1, 9, 14, 29, 1, 30, 1, 1, 0, 6, 5, 33, 4, 11, 5, 14, 3, 3, 18, 9, 9, 15, 4, 17, 2, 27, 10, 41, 2, 2, 7, 11, 5, 4, 5, 4, 4, 3, 9, 14

Offset: 1

Néstor Romeral Andrés, Aug 04 2006

			a(35) = A119957(35) / 35 = 175 / 35 = 5.

Andrei Zabolotskii, Table of n, a(n) for n = 1..8192

Cf. A119957, A007733, A038553. The positions of 1's are A175332.

a[n_] := Total[PowerMod[2, Range[n-MultiplicativeOrder[2, n/2^IntegerExponent[n, 2]], n-1], n]] / n;
Table[a[n], {n,95}] (* Andrei Zabolotskii, Jul 28 2025 *)

Offset corrected by Andrei Zabolotskii, Jul 28 2025

A135547 Number of cycles for vectors of length n under the Ducci map.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 10, 1, 6, 10, 4, 24, 6, 298, 1096, 1, 260, 526, 28, 1098, 16660, 1540, 2050, 2744, 658, 10246, 4870, 599338, 566, 8948416, 34636834, 1, 4198408, 8421892, 4195604, 17043806, 21256, 3538972, 67158052, 35791946, 26214476, 8726292328, 805355524, 806094340, 4296053112, 4297066498, 8388610, 89479864

Offset: 1

N. J. A. Sloane, Feb 24 2008

nonn

The number of different lengths of cycles is given by A111944 and the maximum length by A038553.
Also, the number of connected components in Arnold's graph G_n associated with the Ducci map. G_n has 2^n vertices, one for each binary vector x. Each node x has a single directed edge which goes from x to y, where y_1 = x_2-x_1, y_2 = x_3-x_2, ..., y_{n-1} = x_n-x_{n-1}, y_n = x_1-x_n. (Since the vectors are binary, we could use here sums instead of differences.)
Remarkably, a(n) = A083843(n) for n=4, 7, 8, 14, 16, 23, 28, 31, 32. - Max Alekseyev, Oct 11 2013

Max Alekseyev, Table of n, a(n) for n = 1..420
V. I. Arnold, Complexity of finite sequences of zeros and ones and geometry of finite spaces of functions, Funct. Anal. Other. Math 1(1), 2006, pp. 1-15.
N. J. Calkin, J. G. Stevens, D. M. Thomas, A characterization for the lengths of cycles of the n-number Ducci game, Fib. Q. 43(1), 2005, 53-59.

Cf. A038553, A111944

a(13)-a(19) from R. J. Mathar, Apr 17 2008

Terms a(20) onward added by Max Alekseyev, Oct 12 2013

A138004 Odd numbers k for which all nontrivial cycles of the Ducci map on k-tuples modulo 2 have the same length.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 101, 103, 107, 127, 131, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 233, 239, 251, 263, 269, 271, 281, 283, 293, 307, 311, 313, 317, 347, 349, 359, 367, 373, 379, 383, 389

Offset: 1

T. D. Noe, Feb 26 2008

nonn

All nontrivial cycles have the same length when either k is a prime number with primitive root 2 (see A001122), or when all factors of polynomial ((x+1)^k+1)/x (mod 2) have the same multiplicative order. The corresponding cycle lengths are listed in A138006.
It is conjectured that all terms of this sequence are prime numbers.
Numbers k such that A111944(k) = 2. - Max Alekseyev, Jul 10 2025

Max Alekseyev, Table of n, a(n) for n = 1..310
Florian Breuer, Ernest Lotter and Brink van der Merve, Ducci sequences and cyclotomic polynomials, Finite Fields Appl. 13 (2007), 293-304.
Michal Misiurewicz, John G. Stevens and Diana M. Thomas, Iterations of Linear Maps over Finite Fields, Linear Algebra and its Applications, Vol. 413 (2006), 218-234.

Cf. A001122, A038553, A111944, A138005, A138006.

Edited and terms a(36) onward added by Max Alekseyev, Jul 10 2025

A138006 Length of all the nontrivial cycles of the Ducci map modulo 2 for the dimensions in A138004.

Original entry on oeis.org

3, 15, 7, 341, 819, 9709, 2047, 475107, 31, 3233097, 5461, 8388607, 3556769739, 31675383749, 65498251203, 575525617597, 34359738367, 549755813887, 182518930210733, 2047, 37905296863701641, 2251799813685247, 963770320257286037, 127, 4833046947311902523261, 82051117639860085587829, 2814530423790308547362667

Offset: 1

T. D. Noe, Feb 26 2008

nonn

Max Alekseyev, Table of n, a(n) for n = 1..292

Subsequence of A038553.

C.f. A135546, A138004.

a(n) = A038553(A138004(n)). - Max Alekseyev, Jul 10 2025

Edited and terms a(23) onward added by Max Alekseyev, Jul 10 2025

A135546 Let p be the n-th prime and let g be the order of 2 mod p (see A014664). Then if g is even, a(n) = p*(2^(g/2) - 1), otherwise a(n) = 2^g - 1.

Original entry on oeis.org

3, 15, 7, 341, 819, 255, 9709, 2047, 475107, 31, 9699291, 41943, 5461, 8388607, 3556769739, 31675383749, 65498251203, 575525617597, 34359738367, 511, 549755813887, 182518930210733, 2047, 1627389855, 113715890591104923, 2251799813685247, 963770320257286037

Offset: 2

N. J. A. Sloane, Feb 24 2008

nonn

Karpenkov asks how often is it the case that if p is the n-th prime (n >= 2) then A038553(p) = a(n)? The first failure is at p = 37. Is it true that a(n) is always divisible by A038553(p)?

N. J. A. Sloane, Table of n, a(n) for n = 2..1000
O. N. Karpenkov, On examples of difference operators for {0,1}-valued functions over finite sets, Funct. Anal. Other Math. 1 (2006), 175-180. [The function q(n)]
O. N. Karpenkov, On examples of difference operators for {0,1}-valued functions over finite sets, arXiv:math/0611940 [math.CO], 2006.

Cf. A038553, A014664.

(First load the b-file for A014664 as the array b1.)
a := proc(i) local p,g; p:=ithprime(i); g:=b1[i-1]; if g mod 2 = 0 then p*(2^(g/2)-1) else 2^g-1; fi; end;

g[n_]:=MultiplicativeOrder[2, Prime[n]];a[n_]:=If[EvenQ[g[n]],Prime[n]*(2^(g[n]/2)-1),2^g[n]-1];Table[a[n],{n,2,28}] (* James C. McMahon, Apr 16 2025 *)

A138005 Odd prime numbers p for which not all nontrivial cycles of the Ducci map on p-tuples modulo 2 have the same length.

Original entry on oeis.org

17, 41, 73, 97, 109, 113, 137, 193, 229, 241, 257, 277, 331, 337, 353, 397, 433, 449, 571, 577, 631, 641, 673, 683, 733, 769, 953, 997, 1009, 1013, 1021, 1033, 1051, 1069, 1093, 1103, 1153, 1181, 1201, 1217, 1297, 1321, 1361, 1429, 1489, 1597, 1601, 1613, 1657, 1709, 1789, 1801, 1831, 1873, 1889, 1933, 2011, 2017, 2081, 2113, 2137, 2143, 2153, 2161, 2179, 2273, 2297, 2341, 2381, 2441, 2521, 2593, 2617, 2633

Offset: 1

T. D. Noe, Feb 26 2008

nonn

Primes p such that A111944(p) > 2. - Max Alekseyev, Jul 10 2025

Cf. A038553, A111944, A138004.

a(8)-a(74) from Max Alekseyev, Jul 10 2025

cp's OEIS Frontend

A111944 Number of different lengths of cycles for vectors of length n under the Ducci map.

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A136043 Period-lengths of the base-2 MR-expansions of the reciprocals of the positive integers.

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A119513 a(n) = A119957(n) / n.

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A135547 Number of cycles for vectors of length n under the Ducci map.

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A138004 Odd numbers k for which all nontrivial cycles of the Ducci map on k-tuples modulo 2 have the same length.

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A138006 Length of all the nontrivial cycles of the Ducci map modulo 2 for the dimensions in A138004.

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A135546 Let p be the n-th prime and let g be the order of 2 mod p (see A014664). Then if g is even, a(n) = p*(2^(g/2) - 1), otherwise a(n) = 2^g - 1.

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A138005 Odd prime numbers p for which not all nontrivial cycles of the Ducci map on p-tuples modulo 2 have the same length.

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