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 1-10 of 11 results. Next

A023161 Number of cycles of function f(x) = x^10 mod n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 4, 2, 4, 4, 2, 4, 4, 8, 4, 2, 2, 8, 10, 4, 8, 4, 7, 4, 2, 8, 10, 8, 3, 8, 4, 2, 4, 4, 8, 8, 10, 20, 8, 4, 2, 16, 7, 4, 8, 14, 3, 4, 7, 4, 4, 8, 4, 20, 4, 8, 20, 6, 3, 8, 4, 8, 16, 2, 8, 8, 19, 4, 14, 16, 3, 8, 10, 20, 4, 20, 8, 16, 10, 4, 16, 4, 10, 16, 4, 14, 6, 4, 7, 16, 16, 14, 8, 6, 20
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Not multiplicative; the smallest counterexample is a(667). - T. D. Noe, Nov 14 2006

Links

Crossrefs

Cf. A023153
Cf. A023153-A023160 (cycles of the functions f(x)=x^k mod n for k=2..9)

A117988 Number of functions f:[n]->[n] such that f[(x^2) mod n]=[f(x)^2] mod n for all x in [n], for n=1,2,3,... Here [n] denotes {0,1,2,...,n-1}.

Original entry on oeis.org

1, 4, 6, 16, 18, 576, 78, 1728, 1365, 5184, 486, 2985984, 3474, 389376, 13583700, 268435456, 65538, 119246400, 45006, 39261044736, 21400013700, 15116544, 67590, 45476068117708800, 8696104065, 772395264, 19496328075, 1822309056774144, 231340050
Offset: 1

Views

Author

John W. Layman, Apr 14 2006

Keywords

Comments

See A117986 and A117987 for results on other modular functional equations.

Links

Crossrefs

Cf. A023153, A117986, A117987.

Programs

  • PARI
    \\ See Links section.

Extensions

More terms from Rémy Sigrist, Sep 20 2019

A023154 Number of cycles of function f(x) = x^3 mod n.

Original entry on oeis.org

1, 2, 3, 3, 4, 6, 3, 5, 3, 8, 5, 9, 4, 6, 12, 7, 8, 6, 3, 12, 9, 10, 7, 15, 8, 8, 3, 9, 8, 24, 5, 11, 15, 16, 12, 9, 4, 6, 12, 20, 14, 18, 5, 15, 12, 14, 7, 21, 5, 16, 24, 12, 16, 6, 22, 15, 9, 16, 5, 36, 8, 10, 9, 15, 17, 30, 7, 24, 21, 24, 11, 15, 6, 8, 24, 9, 15, 24, 11, 30, 3, 28, 13, 27, 37
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Not multiplicative; the smallest counterexample is a(55). - T. D. Noe, Nov 14 2006

Links

Crossrefs

Cf. A023153
Cf. A023153-A023161 (cycles of the functions f(x)=x^k mod n for k=2..10)

A023160 Number of cycles of function f(x) = x^9 mod n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 3, 5, 3, 10, 7, 9, 5, 6, 15, 9, 13, 6, 3, 15, 9, 14, 7, 15, 13, 10, 3, 9, 13, 30, 7, 17, 21, 26, 15, 9, 5, 6, 15, 25, 25, 18, 7, 21, 15, 14, 7, 27, 7, 26, 39, 15, 21, 6, 35, 15, 9, 26, 7, 45, 13, 14, 9, 25, 25, 42, 7, 39, 21, 30, 19, 15, 9, 10, 39, 9, 21, 30, 11, 45, 3, 50, 23
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Not multiplicative; the smallest counterexample is a(187). - T. D. Noe, Nov 14 2006

Links

Crossrefs

Cf. A023153
Cf. A023153-A023161 (cycles of the functions f(x)=x^k mod n for k=2..10)

A023155 Number of cycles of function f(x) = x^4 mod n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 4, 2, 4, 4, 4, 4, 4, 8, 4, 2, 2, 8, 6, 4, 8, 8, 4, 4, 4, 8, 6, 8, 4, 8, 10, 2, 8, 4, 8, 8, 6, 12, 8, 4, 4, 16, 10, 8, 8, 8, 4, 4, 10, 8, 4, 8, 4, 12, 8, 8, 12, 8, 4, 8, 10, 20, 16, 2, 8, 16, 10, 4, 8, 16, 10, 8, 6, 12, 8, 12, 16, 16, 10, 4, 8, 8, 6, 16, 4, 20, 8, 8, 4, 16, 16, 8, 20, 8, 12
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Not multiplicative; the smallest counterexample is a(275). - T. D. Noe, Nov 14 2006

Links

Crossrefs

Cf. A023153
Cf. A023153-A023161 (cycles of the functions f(x)=x^k mod n for k=2..10)

A023156 Number of cycles of function f(x) = x^5 mod n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 5, 5, 5, 10, 3, 9, 9, 10, 15, 9, 9, 10, 7, 15, 15, 6, 7, 15, 5, 18, 7, 15, 9, 30, 5, 13, 9, 18, 25, 15, 13, 14, 27, 25, 7, 30, 11, 9, 25, 14, 5, 27, 11, 10, 27, 27, 17, 14, 15, 25, 21, 18, 7, 45, 9, 10, 29, 17, 45, 18, 13, 27, 21, 50, 5, 25, 23, 26, 15, 21, 15, 54, 23, 45, 9
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Not multiplicative; the smallest counterexample is a(63). - T. D. Noe, Nov 14 2006

Links

Crossrefs

Cf. A023153
Cf. A023153-A023161 (cycles of the functions f(x)=x^k mod n for k=2..10)

Extensions

Corrected by Charles R Greathouse IV, Sep 02 2009

A023157 Number of cycles of function f(x) = x^6 mod n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 6, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 12, 3, 4, 6, 4, 2, 4, 5, 8, 6, 2, 12, 4, 4, 4, 2, 4, 4, 4, 6, 8, 5, 12, 4, 6, 4, 4, 5, 12, 4, 4, 3, 4, 12, 4, 4, 10, 4, 8, 6, 12, 4, 2, 4, 24, 3, 4, 6, 8, 21, 4, 2, 4, 12, 4, 12, 8, 3, 4, 2, 12, 3, 8, 4, 10, 10, 12, 3, 8, 4, 6, 12, 8, 4, 4, 2, 10, 12
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Not multiplicative; the smallest counterexample is a(667). - T. D. Noe, Nov 14 2006

Links

Crossrefs

Cf. A023153-A023161 (cycles of the functions f(x)=x^k mod n for k=2..10).

A023158 Number of cycles of function f(x) = x^7 mod n.

Original entry on oeis.org

1, 2, 3, 3, 4, 6, 7, 5, 7, 8, 5, 9, 10, 14, 12, 7, 10, 14, 11, 12, 21, 10, 5, 15, 8, 20, 11, 21, 4, 24, 13, 11, 15, 20, 28, 21, 16, 22, 30, 20, 14, 42, 7, 15, 28, 10, 5, 21, 7, 16, 30, 30, 8, 22, 22, 35, 33, 8, 11, 36, 22, 26, 49, 19, 43, 30, 13, 30, 15, 56, 5, 35, 26, 32, 24, 33, 35, 60
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Not multiplicative; the smallest counterexample is a(55). - T. D. Noe, Nov 14 2006

Links

Crossrefs

Cf. A023153
Cf. A023153-A023161 (cycles of the functions f(x)=x^k mod n for k=2..10)

A023159 Number of cycles of function f(x) = x^8 mod n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 3, 2, 3, 4, 3, 4, 3, 6, 4, 2, 2, 6, 6, 4, 6, 6, 3, 4, 3, 6, 6, 6, 8, 8, 6, 2, 6, 4, 6, 6, 6, 12, 6, 4, 3, 12, 15, 6, 6, 6, 4, 4, 15, 6, 4, 6, 5, 12, 6, 6, 12, 16, 3, 8, 6, 12, 10, 2, 6, 12, 6, 4, 6, 12, 15, 6, 6, 12, 6, 12, 10, 12, 12, 4, 9, 6, 4, 12, 4, 30, 16, 6, 3, 12, 10, 6, 12, 8, 12, 4
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Not multiplicative; the smallest counterexample is a(63). - T. D. Noe, Nov 14 2006

Links

Crossrefs

Cf. A023153
Cf. A023153-A023161 (cycles of the functions f(x)=x^k mod n for k=2..10)

A323424 Number of cycles (mod n) under Collatz map.

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, Jan 14 2019

Keywords

Comments

This sequence is likely to be unbounded.

Examples

			The initial terms, alongside the corresponding cycles, are:
  n   a(n)  cycles
  --  ----  --------------------
   1     1  (0)
   2     1  (0)
   3     2  (0), (1)
   4     1  (0)
   5     2  (0), (1, 4, 2)
   6     2  (0), (1, 4, 2)
   7     3  (0), (1, 4, 2), (3)
   8     2  (0), (1, 4, 2)
   9     2  (0), (1, 4, 2)
  10     2  (0), (1, 4, 2)
  11     3  (0), (1, 4, 2), (5)
  12     2  (0), (1, 4, 2)
  13     3  (0), (1, 4, 2), (3, 10, 5)
  14     2  (0), (1, 4, 2)
  15     3  (0), (1, 4, 2), (7)
  16     2  (0), (1, 4, 2)
  17     2  (0), (1, 4, 2)
  18     2  (0), (1, 4, 2)
  19     3  (0), (1, 4, 2), (9)
  20     2  (0), (1, 4, 2)

Links

Crossrefs

See A000374, A023135, A023153, A233521 for similar sequences.
Cf. A006370.

Programs

  • PARI
    a(n, f = k -> if (k%2, 3*k+1, k/2)) = { my (c=0, s=0); for (k=0, n-1, if (!bittest(s, k), my (v=0, i=k); while (1, v += 2^i; i = f(i) % n; if (bittest(s, i), break, bittest(v, i), c++; break)); s += v)); return (c) }

Formula

a(n) >= 2 for any n > 4 (as we have at least the cycles (0) and (1, 4, 2)).
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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