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 1-6 of 6 results.

A054708 Number of powers of 7 modulo n.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 2, 2, 3, 4, 10, 2, 12, 2, 4, 2, 16, 3, 3, 4, 2, 10, 22, 2, 4, 12, 9, 3, 7, 4, 15, 4, 10, 16, 5, 6, 9, 3, 12, 4, 40, 2, 6, 10, 12, 22, 23, 2, 3, 4, 16, 12, 26, 9, 20, 3, 3, 7, 29, 4, 60, 15, 4, 8, 12, 10, 66, 16, 22, 5, 70, 6, 24, 9, 4, 6, 11, 12, 78, 4, 27, 40, 41, 3, 16, 6
Offset: 1

Views

Author

Henry Bottomley, Apr 20 2000

Keywords

Links

Crossrefs

Cf. A007738, A214411.
Cf. A054703 (base 2), A054704 (3), A054705 (4), A054706 (5), A054707 (6), A054709 (8), A054717 (9), A054710 (10), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

Programs

  • Mathematica
    a[n_] := Module[{e = IntegerExponent[n, 7]}, e + MultiplicativeOrder[7, n/7^e]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

Formula

a(n) = A007738(n) + A214411(n). - Amiram Eldar, Aug 25 2024

A007737 Period of repeating digits of 1/n in base 6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 10, 1, 12, 2, 1, 1, 16, 1, 9, 1, 2, 10, 11, 1, 5, 12, 1, 2, 14, 1, 6, 1, 10, 16, 2, 1, 4, 9, 12, 1, 40, 2, 3, 10, 1, 11, 23, 1, 14, 5, 16, 12, 26, 1, 10, 2, 9, 14, 58, 1, 60, 6, 2, 1, 12, 10, 33, 16, 11, 2, 35, 1, 36, 4, 5, 9, 10, 12, 78, 1, 1, 40, 82, 2, 16, 3, 14, 10
Offset: 1

Views

Author

N. J. A. Sloane, Hal Sampson (hals(AT)easynet.com)

Keywords

Comments

Not multiplicative. Smallest counterexample: a(77)=10, but a(7) = 2 and a(11) = 10. - Mitch Harris, May 16 2005.

Links

Crossrefs

Cf. A007733 (base 2), A007734 (3), A007735 (4), A007736 (5), A007738 (7), A007739 (8), A007740 (9), A007732 (10).

Programs

  • Mathematica
    DigitCycleLength[r_Rational, b_Integer?Positive] := MultiplicativeOrder[b, FixedPoint[ Quotient[#, GCD[#, b]] &, Denominator[r]]]; DigitCycleLength[1, b_Integer?Positive] = 1; Array[ DigitCycleLength[1/#, 6] &, 80] (* Robert G. Wilson v, Jun 10 2011 *)
a[n_] := MultiplicativeOrder[6, n/Times @@ ({2, 3}^IntegerExponent[n, {2, 3}])]; Array[a, 100] (* Amiram Eldar, Aug 26 2024 *)
  • PARI
    a(n)=znorder(Mod(6, n/2^valuation(n, 2)/3^valuation(n, 3))); \\ Joerg Arndt, Dec 14 2014

Extensions

More terms from David W. Wilson

A066799 Square array read by antidiagonals of eventual period of powers of k mod n; period of repeating digits of 1/n in base k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 2, 1, 4, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 6, 1, 1, 1, 1, 1, 4, 1, 6, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 2, 2, 3, 4, 10, 1, 1, 1, 2, 1, 2, 2, 1, 1, 6, 2, 5, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 5, 2, 12
Offset: 1

Views

Author

Henry Bottomley, Dec 20 2001

Keywords

Comments

The determinant of the n X n matrix made from the northwest corner of this array is 0^(n-1). - Iain Fox, Mar 12 2018

Examples

			Rows start: 1,1,1,1,1,...; 1,1,1,1,1,...; 1,2,1,1,2,...; 1,1,2,1,1; 1,4,4,2,1,... T(3,2)=2 since the powers of 2 become 1,2,1,2,1,2,... mod 3 with period 2. T(4,2)=1 since the powers of 2 become 1,2,0,0,0,0,... mod 4 with eventual period 1.
Beginning of array:
+-----+--------------------------------------------------------------------
| n\k |  1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  ...
+-----+--------------------------------------------------------------------
|  1  |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
|  2  |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
|  3  |  1,  2,  1,  1,  2,  1,  1,  2,  1,  1,  2,  1,  1,  2,  1,  1, ...
|  4  |  1,  1,  2,  1,  1,  1,  2,  1,  1,  1,  2,  1,  1,  1,  2,  1, ...
|  5  |  1,  4,  4,  2,  1,  1,  4,  4,  2,  1,  1,  4,  4,  2,  1,  1, ...
|  6  |  1,  2,  1,  1,  2,  1,  1,  2,  1,  1,  2,  1,  1,  2,  1,  1, ...
|  7  |  1,  3,  6,  3,  6,  2,  1,  1,  3,  6,  3,  6,  2,  1,  1,  3, ...
|  8  |  1,  1,  2,  1,  2,  1,  2,  1,  1,  1,  2,  1,  2,  1,  2,  1, ...
| ... |

Links

Crossrefs

Columns are A000012, A007733, A007734, A007735, A007736, A007737, A007738, A007739, A007740, A007732. A002322 is the highest value in each row and the least common multiple of each row, while the number of distinct values in each row is A066800.

Programs

  • Mathematica
    t[n_, k_] := For[p = PowerMod[k, n, n]; m = n + 1, True, m++, If[PowerMod[k, m, n] == p, Return[m - n]]]; Flatten[Table[t[n - k + 1, k], {n, 1, 14}, {k, n, 1, -1}]] (* Jean-François Alcover, Jun 04 2012 *)
  • PARI
    a(n, k) = my(p=k^n%n); for(m=n+1, +oo, if(k^m%n==p, return(m-n))) \\ Iain Fox, Mar 12 2018

Formula

T(n, k) = T(n, k-n) if k > n.
T(n, n) = T(n, n+1) = 1.
T(n, n-1) = 2.

A007735 Period of base 4 representation of 1/n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 5, 1, 6, 3, 2, 1, 4, 3, 9, 2, 3, 5, 11, 1, 10, 6, 9, 3, 14, 2, 5, 1, 5, 4, 6, 3, 18, 9, 6, 2, 10, 3, 7, 5, 6, 11, 23, 1, 21, 10, 4, 6, 26, 9, 10, 3, 9, 14, 29, 2, 30, 5, 3, 1, 6, 5, 33, 4, 11, 6, 35, 3, 9, 18, 10, 9, 15, 6, 39, 2, 27, 10, 41, 3, 4, 7, 14, 5, 11, 6, 6, 11, 5
Offset: 1

Views

Author

N. J. A. Sloane, Hal Sampson (hals(AT)easynet.com)

Keywords

Links

Crossrefs

Cf. A007733 (base 2), A007734 (3), A007736 (5), A007737 (6), A007738 (7), A007739 (8), A007740 (9), A007732 (10).

Programs

  • Mathematica
    DigitCycleLength[r_Rational, b_Integer?Positive] := MultiplicativeOrder[b, FixedPoint[ Quotient[#, GCD[#, b]] &, Denominator[r]]]; DigitCycleLength[1, b_Integer?Positive] = 1; Array[ DigitCycleLength[1/#, 4] &, 80] (* Robert G. Wilson v, Jun 10 2011 *)
a[n_] := MultiplicativeOrder[4, n/2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Aug 26 2024 *)

Extensions

More terms from David W. Wilson

A007740 Period of repeating digits of 1/n in base 9.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 5, 1, 3, 3, 2, 2, 8, 1, 9, 2, 3, 5, 11, 1, 10, 3, 1, 3, 14, 2, 15, 4, 5, 8, 6, 1, 9, 9, 3, 2, 4, 3, 21, 5, 2, 11, 23, 2, 21, 10, 8, 3, 26, 1, 10, 3, 9, 14, 29, 2, 5, 15, 3, 8, 6, 5, 11, 8, 11, 6, 35, 1, 6, 9, 10, 9, 15, 3, 39, 2, 1, 4, 41, 3, 8, 21, 14, 5, 44, 2, 3, 11, 15, 23
Offset: 1

Views

Author

N. J. A. Sloane, Hal Sampson (hals(AT)easynet.com)

Keywords

Comments

Not multiplicative. Smallest counterexample a(5) = a(16) = 2, but a(80) = 2. - David W. Wilson, Jun 09 2005

Links

Crossrefs

Cf. A007733 (base 2), A007734 (3), A007735 (4), A007736 (5), A007737 (6), A007738 (7), A007739 (8), A007732 (10).

Programs

  • Mathematica
    DigitCycleLength[r_Rational, b_Integer?Positive] := MultiplicativeOrder[b, FixedPoint[ Quotient[#, GCD[#, b]] &, Denominator[r]]]; DigitCycleLength[1, b_Integer?Positive] = 1; Array[ DigitCycleLength[1/#, 9] &, 80] (* Robert G. Wilson v, Jun 10 2011 *)
a[n_] := MultiplicativeOrder[9, n/3^IntegerExponent[n, 3]]; Array[a, 100] (* Amiram Eldar, Aug 26 2024 *)

A007739 Period of repeating digits of 1/n in base 8.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 1, 1, 2, 4, 10, 2, 4, 1, 4, 1, 8, 2, 6, 4, 2, 10, 11, 2, 20, 4, 6, 1, 28, 4, 5, 1, 10, 8, 4, 2, 12, 6, 4, 4, 20, 2, 14, 10, 4, 11, 23, 2, 7, 20, 8, 4, 52, 6, 20, 1, 6, 28, 58, 4, 20, 5, 2, 1, 4, 10, 22, 8, 22, 4, 35, 2, 3, 12, 20, 6, 10, 4, 13, 4, 18, 20, 82, 2, 8, 14, 28, 10, 11, 4, 4
Offset: 1

Views

Author

N. J. A. Sloane, Hal Sampson (hals(AT)easynet.com)

Keywords

Links

Crossrefs

Cf. A007733 (base 2), A007734 (3), A007735 (4), A007736 (5), A007737 (6), A007738 (7), A007740 (9), A007732 (10).

Programs

  • Mathematica
    DigitCycleLength[r_Rational, b_Integer?Positive] := MultiplicativeOrder[b, FixedPoint[ Quotient[#, GCD[#, b]] &, Denominator[r]]]; DigitCycleLength[1, b_Integer?Positive] = 1; Array[ DigitCycleLength[1/#, 8] &, 80] (* Robert G. Wilson v, Jun 10 2011 *)
a[n_] := MultiplicativeOrder[8, n/2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Aug 26 2024 *)

Extensions

More terms from David W. Wilson
Showing 1-6 of 6 results.

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

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