A007734 -id:A007734 - cp's OEIS frontend

A054704 Number of powers of 3 modulo n.

1, 1, 2, 2, 4, 2, 6, 2, 3, 4, 5, 3, 3, 6, 5, 4, 16, 3, 18, 4, 7, 5, 11, 3, 20, 3, 4, 6, 28, 5, 30, 8, 6, 16, 12, 4, 18, 18, 4, 4, 8, 7, 42, 10, 6, 11, 23, 5, 42, 20, 17, 6, 52, 4, 20, 6, 19, 28, 29, 5, 10, 30, 8, 16, 12, 6, 22, 16, 12, 12, 35, 4, 12, 18, 21, 18, 30, 4, 78, 4, 5, 8, 41, 7, 16

Offset: 1

Henry Bottomley, Apr 20 2000

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from David W. Wilson)

Cf. A007734, A007949.

Cf. A054703 (base 2), A054705 (4), A054706 (5), A054707 (6), A054708 (7), A054709 (8), A054717 (9), A054710 (10), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

a[n_] := Module[{e = IntegerExponent[n, 3]}, e + MultiplicativeOrder[3, n/3^e]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = A007734(n) + A007949(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

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

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

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

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 *)

a(n)=znorder(Mod(6, n/2^valuation(n, 2)/3^valuation(n, 3))); \\ Joerg Arndt, Dec 14 2014

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

Henry Bottomley, Dec 20 2001

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

			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, ...
| ... |

Iain Fox, Antidiagonals n = 1..141 of array, flattened

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.

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 *)

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

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

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

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

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 *)

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

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

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

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

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

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

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

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 *)

More terms from David W. Wilson

A173934 Irregular triangle in which row n consists of numbers m < k/2 such that m/k is in the Cantor set, where k= A173931(n) and gcd(m,k) = 1.

Original entry on oeis.org

1, 1, 3, 1, 3, 4, 1, 3, 9, 1, 3, 9, 13, 1, 3, 7, 9, 19, 21, 25, 27, 1, 3, 9, 10, 27, 30, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 5, 11, 15, 33, 45, 47, 5, 15, 41, 45, 47, 59, 7, 16, 21, 22, 48, 61, 63, 66, 1, 3, 7, 9, 19, 21, 25, 27, 55, 57, 63, 73, 75, 79, 81, 1, 3, 9, 27

Offset: 1

T. D. Noe, Mar 03 2010

The length of row n is A173933(n). Observe that the m are actually less than k/3. Note that (k-m)/k is also in the Cantor set. If m appears in a row, then 3m does also. Let A and B be the first and last numbers in row n, then it appears that k = A + 3B. This implies A = k (mod 3). The interesting graph of this triangle shows that some ranges of m are not allowed.

When k is a prime of the form (3^r-1)/2, then the row consists of the 2^(r-1)-1 numbers (greater than 0) whose base-3 representation consists of only 0's and 1's. Hence, for r=3,7, and 13, the primes k are 13, 1093, and 797161, and the number of m < k/2 is 3, 63, and 4095.

T. D. Noe, Rows n=1..185, flattened

Cf. A005836, A007734, A054591, A173931, A173933, A191106, A306556.

Flatten[Last[Transpose[cantor]]] (* see A173931 *)

Name qualified by Peter Munn, Jul 06 2019

A333334 a(n) is the smallest positive number k such that n divides 3^k + k.

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 6, 5, 9, 3, 2, 9, 10, 15, 3, 13, 4, 9, 18, 17, 6, 29, 22, 21, 23, 17, 27, 25, 28, 3, 5, 13, 57, 23, 6, 9, 36, 23, 12, 37, 40, 15, 17, 29, 63, 63, 35, 45, 6, 23, 27, 17, 19, 27, 57, 109, 18, 31, 10, 57, 52, 5, 90, 45, 17, 57, 66, 65, 63, 23, 70

Offset: 1

Jinyuan Wang, Mar 15 2020

For any positive integer n, if k = a(n) + n*m*A007734(n) and m >= 0 then 3^k + k is divisible by n.

Seiichi Manyama, Table of n, a(n) for n = 1..5000
Brazil National Olympiad, 2005, Problem 6

Cf. A007734, A247248, A333335, A333336, A333339.

a[n_] := Module[{k = 1}, While[!Divisible[3^k + k, n], k++]; k]; Array[a, 100] (* Amiram Eldar, Mar 16 2020 *)

a(n) = for(k=1, oo, if(Mod(3, n)^k==-k, return(k)));

a(3^m) = 3^m for m >= 0.

a(p) <= p - 1 if p is a prime greater than 3.

A333339 a(n) is the smallest positive number k such that n divides 3^k - k.

Original entry on oeis.org

1, 1, 3, 3, 7, 3, 2, 3, 9, 7, 4, 3, 16, 5, 27, 11, 5, 9, 29, 7, 27, 45, 39, 3, 73, 27, 27, 27, 22, 27, 132, 27, 36, 5, 27, 27, 65, 29, 27, 27, 27, 27, 10, 59, 27, 39, 12, 27, 47, 73, 42, 27, 68, 27, 36, 27, 30, 47, 154, 27, 192, 147, 27, 59, 16, 45, 119, 75, 39

Offset: 1

Jinyuan Wang, Mar 16 2020

nonn

For any positive integer n, if k = a(n) + n*m*A007734(n) and m >= 0 then 3^k - k is divisible by n.

a(n) > log_3(n). - Robert Israel, Mar 19 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007734, A072872, A333334.

f:= proc(n) local k;
  for k from 1 do if 3 &^k - k mod n = 0 then return k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Mar 19 2020

a[n_] := Module[{k = 1}, While[!Divisible[3^k - k, n], k++]; k]; Array[a, 100] (* Amiram Eldar, Mar 16 2020 *)

a(n) = for(k=1, oo, if(Mod(3, n)^k==k, return(k)));

a(3^m) = 3^m for m >= 0.

a(3^m-m) = m for m >= 1. - Robert Israel, Mar 19 2020

A336504 3-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_3[x].

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 26, 27, 30, 32, 36, 39, 40, 42, 44, 45, 48, 52, 54, 56, 60, 63, 64, 66, 72, 78, 80, 81, 84, 88, 90, 96, 99, 100, 104, 105, 108, 112, 117, 120, 126, 128, 130, 132, 135, 140, 144, 150, 156, 160, 162, 165, 168, 176, 180

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

For a rational prime number p, a "p-practical number" is a number m such that the polynomial x^m - 1 has a divisor of every degree <= m in F_p[x], the prime field of order p.
A number m is 3-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} A007734(d) * n_d, where A007734(d) is the multiplicative order of 3 modulo the largest divisor of d not divisible by 3, and 0 <= n_d <= phi(d)/A007734(d).
The number of terms not exceeding 10^k for k = 1, 2, ... are 7, 41, 258, 1881, 15069, 127350, 1080749, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack and Lola Thompson, On the degrees of divisors of T^n-1>, New York Journal of Mathematics, Vo. 19 (2013), pp. 91-116, preprint, arXiv:1206.2084 [math.NT], 2012.
Lola Thompson, Products of distinct cyclotomic polynomials, Ph.D. thesis, Dartmouth College, 2012.
Lola Thompson, On the divisors of x^n - 1 in F_p[x], International Journal of Number Theory, Vol. 9, No. 2 (2013), pp. 421-430.
Lola Thompson, Variations on a question concerning the degrees of divisors of x^n - 1, Journal de Théorie des Nombres de Bordeaux, Vol. 26, No. 1 (2014), pp. 253-267.
Eric Weisstein's World of Mathematics, Finite Field.
Wikipedia, Finite field.

Cf. A038502, A007734, A007949.

Cf. A000010, A260653, A336503, A336505.

rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; mo[n_, p_] := MultiplicativeOrder[p, n/p^IntegerExponent[n, p]]; ppQ[n_, p_] := Module[{d = Divisors[n]}, m = mo[#, p] & /@ d; ns = EulerPhi[d]/m; r = rep[m, ns]; Min @ Rest @ CoefficientList[Series[Product[1 + x^r[[i]], {i, Length[r]}], {x, 0, n}], x] >  0]; Select[Range[200], ppQ[#, 3] &]

cp's OEIS Frontend

A054704 Number of powers of 3 modulo n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A007735 Period of base 4 representation of 1/n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A173934 Irregular triangle in which row n consists of numbers m < k/2 such that m/k is in the Cantor set, where k= A173931(n) and gcd(m,k) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A333334 a(n) is the smallest positive number k such that n divides 3^k + k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI