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 1-5 of 5 results.

A038553 Maximum cycle length in differentiation digraph for n-bit binary sequences.

Original entry on oeis.org

1, 1, 3, 1, 15, 6, 7, 1, 63, 30, 341, 12, 819, 14, 15, 1, 255, 126, 9709, 60, 63, 682, 2047, 24, 25575, 1638, 13797, 28, 475107, 30, 31, 1, 1023, 510, 4095, 252, 3233097, 19418, 4095, 120, 41943, 126, 5461, 1364, 4095, 4094, 8388607, 48, 2097151, 51150, 255, 3276, 3556769739, 27594, 1048575
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Length of longest cycle for vectors of length n under the Ducci map.
Also, the period of polynomial (x+1)^n+1 over GF(2) (cf. A046932). - Max Alekseyev, Oct 12 2013
Per the comment by T. D. Noe originally given in A138006, it appears that for an odd n > 1, a(n) <= n*(2^((n-1)/2)-1). - Max Alekseyev, Jul 10 2025

References

  • Simmons, G. J., The structure of the differentiation digraphs of binary sequences. Ars Combin. 35 (1993), A, 71-88. Math. Rev. 95f:05052.

Links

Crossrefs

Cf. A111944, A135547, A136043, A119513.

Formula

It appears that whenever b(n) = log2(a(n)/n + 1) is an integer and n > 1, b(n) = A119513(n) = A136043(n). - Andrei Zabolotskii, Jul 28 2025

Extensions

Entry revised by N. J. A. Sloane, Jun 19 2006, Feb 24 2008
a(46) corrected, terms a(51) onward and b-file added by Max Alekseyev, Oct 12 2013
b-file extended by Max Alekseyev, Sep 24 2019

A136042 Base-2 MR-expansion of 1/29.

Original entry on oeis.org

5, 4, 1, 2, 3, 1, 1, 1, 2, 1, 1, 3, 2, 1, 5, 4, 1, 2, 3, 1, 1, 1, 2, 1, 1, 3, 2, 1, 5, 4, 1, 2, 3, 1, 1, 1, 2, 1, 1, 3, 2, 1, 5, 4, 1, 2, 3, 1, 1, 1, 2, 1, 1, 3, 2, 1, 5, 4, 1, 2, 3, 1, 1, 1, 2, 1, 1, 3, 2, 1, 5, 4, 1, 2, 3, 1, 1, 1, 2, 1, 1, 3, 2, 1, 5, 4, 1, 2, 3, 1, 1, 1, 2, 1, 1, 3, 2, 1, 5, 4, 1, 2, 3, 1, 1
Offset: 1

Views

Author

John W. Layman, Dec 12 2007

Keywords

Comments

The base-m MR-expansion of a positive real number x, denoted by MR(x,m), is the integer sequence {s(1),s(2),s(3),...}, where s(i) is the smallest exponent d such that (m^d)x(i)>1 and where x(i+1)=(m^d)x(i)-1, with the initialization x(1)=x. The base-2 MR-expansion of 1/29 is periodic with period length 14. Further computational results (see A136043) suggest that if p is a prime with 2 as a primitive root, then the base-2 MR-expansion of 1/p is periodic with period (p-1)/2. This has been confirmed for primes up to 2000. The base-2 MR-expansion of e=2.71828... is given in A136044.

Examples

			The MR-expansion of 1/5 using m=2 is {3,1,3,1,3,1,3,1,...}, because 1/5->2/5->4/5->8/5->3/5->6/5->1/5->... indicating that MR(1/5,2) begins {3,1,...} and has period length 2.

Links

Crossrefs

Cf. A136043, A136044, A158379.

A155072 Positive integers n such that the base-2 MR-expansion of 1/n is periodic with period (n-1)/4.

Original entry on oeis.org

17, 41, 97, 137, 193, 313, 401, 409, 449, 521, 569, 761, 769, 809, 857, 929, 977, 1009, 1129, 1297, 1361, 1409, 1489, 1697, 1873, 1993, 2081, 2137, 2153, 2161, 2297, 2377, 2417, 2521, 2609, 2617, 2633, 2713, 2729, 2753, 2777, 2801, 2897, 3001
Offset: 1

Views

Author

John W. Layman, Jan 19 2009

Keywords

Comments

See A136042 for the definition of the MR-expansion of a positive real number.
It appears that all terms of this sequence are primes of the form 8n+1 (A007519).
Apparently a subsequence of A115591. - Mia Boudreau, Jun 17 2025

Examples

			Applying the definition of the base-2 MR-expansion to 1/17 gives 1/17 -> 2/17 -> 4/17 -> 8/17 -> 16/17 -> 32/17 -> 15/17 -> 30/17 -> 13/17 -> 26/17 -> 9/17 -> 18/17 -> 1/17 -> ..., which shows that the expansion begins {5,1,1,1,...} and has period 4=(17-1)/4.

Links

Crossrefs

Cf. A136042, A007519, A115591, A136043.

Programs

  • Mathematica
    a[p_] := 1 + Sum[2 Cos[2^n Pi/((2 p + 1) )], {n, 1, p}];
Select[Range[500], Reduce[a[#]^2 == 2 # + 1, Integers] &];
2 % + 1 (* Gerry Martens, May 01 2016 *)

Extensions

More terms from Mia Boudreau, Jun 17 2025

A136044 Base-2 MR-expansion of e = 2.71828... (the base of natural logarithms).

Original entry on oeis.org

0, 0, 1, 2, 1, 2, 1, 1, 1, 1, 1, 5, 2, 2, 4, 2, 1, 4, 2, 4, 2, 2, 1, 1, 2, 1, 2, 3, 2, 2, 3, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 4, 2, 2, 1, 4, 8, 3, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 2, 1, 4, 2, 1, 1, 3, 1, 1, 4, 2, 1, 6, 1, 1, 1, 3, 1, 1, 4, 2, 1, 2, 3, 1, 2, 1, 2, 3, 2, 2, 1, 2, 2, 3, 1
Offset: 1

Views

Author

John W. Layman, Dec 12 2007

Keywords

Comments

See A136042 for the definition of the MR-expansion of a positive real number.

Examples

			See A136042 for an example of a base-2 MR-expansion.

Crossrefs

Cf. A136042, A136043.

A158379 Period-lengths of the base-3 MR-expansions of the reciprocals of the positive integers.

Original entry on oeis.org

2, 1, 2, 2, 4, 1, 6, 1, 2, 4, 4, 2, 2, 6, 4, 3, 16, 1, 18, 2, 6, 3, 8, 1, 20, 1, 2, 6, 28, 4, 30, 7, 4, 16, 10, 2, 18, 18, 2, 2, 8, 6, 42, 8, 4, 11, 18, 3, 42, 20, 16, 4, 52, 1, 20, 3, 18, 28, 26, 2, 10, 30, 6, 15, 10, 3, 22, 12, 8, 8, 28, 1, 12, 18, 20, 18, 28, 1, 78, 1, 2, 8, 38, 6, 14, 42, 28
Offset: 1

Views

Author

John W. Layman, Mar 17 2009

Keywords

Comments

See A136042 for the definition of the MR-expansion.
It appears that if p is a prime with 3 as a primitive root (A001122), then the MR-expansion of 1/p is periodic with period p-1.
The period lengths of the base-2 MR-expansions of the reciprocals of the positive integers are given in A136043.

Examples

			The base-3 MR-expansion of 1/5 is {2,1,0,1,2,1,0,1,...} because 1/5->3/5->9/5->4/5->12/5->7/5->2/5->6/5->1/5->..., indicating that MR(1/5,3) begins {2,1,0,1,...} and has period 4. Thus a(5)=4.

Crossrefs

Cf. A001122, A007733, A115591, A136042, A136043, A136044, A155072.
Showing 1-5 of 5 results.

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