A036117 -id:A036117 - cp's OEIS frontend

A201908 Irregular triangle of 2^k mod (2n-1).

0, 1, 2, 1, 2, 4, 3, 1, 2, 4, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, 2, 4, 8, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 11, 1, 2, 4, 8, 16, 9

Offset: 1

T. D. Noe, Dec 07 2011

The length of the rows is given by A002326. For n > 1, the first term of row n is 1 and the last term is n. Many sequences are in this one: starting at A036117 (mod 11) and A070335 (mod 23).

Row n, for n >= 2, divided elementwise by (2*n-1) gives the cycles of iterations of the doubling function D(x) = 2*x or 2*x-1 if 0 <= x < 1/2 or , 1/2 <= x < 1, respectively, with seed 1/(2*n-1). See the Devaney reference, pp. 25-26. D^[k](x) = frac(2^k/(2*n-1)), for k = 0, 1, ..., A002326(n-1) - 1. E.g., n = 3: 1/5, 2/5, 4/5, 3/5. - Gary W. Adamson and Wolfdieter Lang, Jul 29 2020.

			The irregular triangle T(n, k) begins:
n\k  0 1 2 3  4  5  6  7 8  9 10 11 12 13 14 15 16 17 ...
---------------------------------------------------------
1:   0
2:   1 2
3:   1 2 4 3
4:   1 2 4
5:   1 2 4 8  7  5
6:   1 2 4 8  5 10  9  7 3  6
7:   1 2 4 8  3  6 12 11 9  5 10  7
8:   1 2 4 8
9:   1 2 4 8 16 15 13  9
10:  1 2 4 8 16 13  7 14 9 18 17 15 11  3  6 12  5 10
... reformatted by _Wolfdieter Lang_, Jul 29 2020.

Robert L. Devaney, A First Course in Chaotic Dynamical Systems, Addison-Wesley., 1992. pp. 24-25

T. D. Noe, Rows n = 1..100, flattened

Cf. A002326, A201909 (3^k), A201910 (5^k), A201911 (7^k).

Cf. A000034 (3), A070402 (5), A069705 (7), A036117 (11), A036118 (13), A062116 (17), A036120 (19), A070347 (21), A070335 (23), A070336 (25), A070337 (27), A036122 (29), A070338 (33), A070339 (35), A036124 (37), A070340 (39), A070348 (41), A070349 (43), A070350 (45), A070351 (47), A036128 (53), A036129 (59), A036130 (61), A036131 (67), A036135 (83), A036138 (101), A036140 (107), A201920 (125), A036144 (131), A036146 (139), A036147 (149), A036150 (163), A036152 (173), A036153 (179), A036154 (181), A036157 (197), A036159 (211), A036161 (227).

R:=List([0..72],n->OrderMod(2,2*n+1));;
Flat(Concatenation([0],List([2..11],n->List([0..R[n]-1],k->PowerMod(2,k,2*n-1))))); # Muniru A Asiru, Feb 02 2019

nn = 30; p = 2; t = p^Range[0, nn]; Flatten[Table[If[IntegerQ[Log[p, n]], {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, 1, nn, 2}]]

T(n, k) = 2^k mod (2*n-1), n >= 1, k = 0, 1, ..., A002326(n-1) - 1.

T(n, k) = (2*n-1)*frac(2^k/(2*n-1)), n >= 1, k = 0, 1, ..., A002326(n-1) - 1, with the fractional part frac(x) = x - floor(x). - Wolfdieter Lang, Jul 29 2020

A010691 Period 2: repeat (1,10).

Original entry on oeis.org

1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10

Offset: 0

N. J. A. Sloane

Regular continued fraction of (5+sqrt 35)/10. - R. J. Mathar, Nov 21 2011

Sequence is an infinite palindrome in two ways (numbers and English names): ONE, TEN, ONE, TEN, ONE, TEN, ONE, ... . - Eric Angelini, Sep 16 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A036117, A070341, A168429, A070367, A070392, A070404, A048271, A187466.

[10^n mod 11: n in [0..80]]; // Vincenzo Librandi, Aug 24 2011

g:=(1+10*z)/((1-z^2)): gser:=series(g, z=0, 66): seq((coeff(gser, z, n)), n=0..65); # Zerinvary Lajos, Feb 25 2009

PadRight[{},100,{1,10}] (* Harvey P. Dale, Aug 27 2013 *)

a(n) = -9/2*(-1)^n + 11/2.
G.f.: (1+10*z)/(1-z^2). - Zerinvary Lajos, Feb 25 2009
a(n) = 10^n mod 11. - M. F. Hasler, Mar 10 2011
From Nicolas Bělohoubek, Nov 11 2021: (Start)
a(n) = 10/a(n-1). See also A010695.
a(n) = 11 - a(n-1). See also A010712. (End)

A168429 a(n) = 4^n mod 11.

Original entry on oeis.org

1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1, 4, 5, 9, 3

Offset: 0

Zerinvary Lajos, Nov 25 2009

Period 5: repeat [1, 4, 5, 9, 3].

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Joshua Ide and Marc S. Renault, Power Fibonacci Sequences, Fib. Q. 50(2), 2012, 175-179.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

A062116 a(n) = 2^n mod 17.

Original entry on oeis.org

1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 15, 13

Offset: 0

Olivier Gérard, Jun 06 2001

Period 8.

			a(5) = 32 mod 17 = 15.

I. M. Vinogradov, Elements of Number Theory, pp. 220 ff.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,-1,1).

Cf. A036117, A036118.

a:=List([0..70],n->PowerMod(2,n,17));; Print(a); # Muniru A Asiru, Jan 29 2019

[2^n mod 17: n in [0..100]]; // G. C. Greubel, Oct 16 2018

Mod[#,17]&/@(2^Range[0,100])  (* Harvey P. Dale, Mar 06 2011 *)

a(n) = { lift(Mod(2,17)^n) } \\ Harry J. Smith, Aug 01 2009

[power_mod(2,n,17) for n in range(0,87)] # Zerinvary Lajos, Nov 03 2009

From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n-1) - a(n-4) + a(n-5).
G.f.: (1 + x + 2*x^2 + 4*x^3 + 9*x^4)/((1-x)*(1+x^4)). (End)
a(n) = 17 - a(n+4) = a(n+8) for all n in Z. - Michael Somos, Oct 17 2018

A201912 Irregular triangle of 2^k mod prime(n).

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 3, 1, 2, 4, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 9, 18, 13, 3, 6, 12, 1, 2, 4, 8, 16, 3, 6, 12, 24

Offset: 1

T. D. Noe, Dec 17 2011

The row lengths are in A014664. For n > 1, the first term of each row is 1 and the last term is 2*prime(n)-1, which is A006254. Many sequences are in this one.

			The first 11 rows are:
2:  0;
3:  1, 2;
5:  1, 2, 4, 3;
7:  1, 2, 4;
11: 1, 2, 4, 8,  5, 10,  9,  7,  3,  6;
13: 1, 2, 4, 8,  3,  6, 12, 11,  9,  5, 10,  7;
17: 1, 2, 4, 8, 16, 15, 13,  9;
19: 1, 2, 4, 8, 16, 13,  7, 14,  9, 18, 17, 15, 11,  3,  6, 12,  5, 10;
23: 1, 2, 4, 8, 16,  9, 18, 13,  3,  6, 12;
29: 1, 2, 4, 8, 16,  3,  6, 12, 24, 19,  9, 18,  7, 14, 28, 27, 25, 21, 13, 26, 23, 17, 5, 10, 20, 11, 22, 15;
31: 1, 2, 4, 8, 16;

T. D. Noe, Rows n = 1..60, flattened

Cf. A062117, A201908, A201909, A201910, A201911.

Cf. similar sequences of the type 2^n mod p, where p is a prime: A000034 (p=3), A070402 (p=5), A069705 (p=7), A036117 (p=11), A036118 (p=13), A062116 (p=17), A036120 (p=19), A070335 (p=23), A036122 (p=29), A269266 (p=31), A036124 (p=37), A070348 (p=41), A070349 (p=43), A070351 (p=47), A036128 (p=53), A036129 (p=59), A036130 (p=61), A036131 (p=67), A036135 (p=83), A036138 (p=101), A036140 (p=107), A036144 (p=131), A036146 (p=139), A036147 (p=149), A036150 (p=163), A036152 (p=173), A036153 (p=179), A036154 (p=181), A036157 (p=197), A036159 (p=211), A036161 (p=227).

P:=Filtered([1..350],IsPrime);;
R:=List([1..Length(P)],n->OrderMod(2,P[n]));;
Flat(Concatenation([0],List([2..10],n->List([0..R[n]-1],k->PowerMod(2,k,P[n]))))); # Muniru A Asiru, Feb 01 2019

nn = 10; p = 2; t = p^Range[0,Prime[nn]]; Flatten[Table[If[Mod[n, p] == 0, {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, Prime[Range[nn]]}]]

A187466 a(n) = 9^n mod 11.

Original entry on oeis.org

1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1

Offset: 0

M. F. Hasler, Mar 10 2011

Period 5: repeat [1, 9, 4, 3, 5].

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A010691, A036117, A048271, A070341, A070367, A070392, A070404, A168429.

A187466:=n->9^n mod 11: seq(A187466(n), n=0..100); # Wesley Ivan Hurt, Jun 11 2016

PowerMod[9, Range[0, 100], 11] (* Wesley Ivan Hurt, Jun 11 2016 *)

a(n)=lift(Mod(9,11)^n) \\ Charles R Greathouse IV, Mar 22 2016

G.f.: (5*x^4 + 3*x^3 + 4*x^2 + 9*x + 1)/(1 - x^5). - Chai Wah Wu, Jun 04 2016

a(n) = a(n-5) for n>4. - Wesley Ivan Hurt, Jun 11 2016

A269266 a(n) = 2^n mod 31.

Original entry on oeis.org

1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1

Offset: 0

Vincenzo Librandi, Mar 31 2016

Continued fraction expansion of (1651+sqrt(3236405))/2386. - Bruno Berselli, Mar 31 2016

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A201912 (11th row of the triangle).

Cf. similar sequences of the type 2^n mod p, where p is a prime: A000034 (p=3), A070402 (p=5), A069705 (p=7), A036117 (p=11), A036118 (p=13), A062116 (p=17), A036120 (p=19), A070335 (p=23), A036122 (p=29), this sequence (p=31), A036124 (p=37), A070348 (p=41), A070349 (p=43), A070351 (p=47), A036128 (p=53), A036129 (p=59), A036130 (p=61), A036131 (p=67).

List([0..70],n->PowerMod(2,n,31)); # Muniru A Asiru, Jan 30 2019

Magma
```
[Modexp(2, n, 31): n in [0..100]];
```

&cat [[1,2,4,8,16]^^20] // Bruno Berselli, Mar 31 2016

Mathematica
```
PowerMod[2, Range[0, 100], 31]
```

a(n)=2^(n%5) \\ Charles R Greathouse IV, Mar 31 2016

x='x+O('x^99); Vec((1+2*x+4*x^2+8*x^3+16*x^4)/(1-x^5)) \\ Altug Alkan, Mar 31 2016

for n in range(0,100):print(2**n%31) # Soumil Mandal, Apr 03 2016

def A269266(n): return pow(2,n,31) # Chai Wah Wu, Jan 03 2022

[2^mod(n,5) for n in (0..100)] # Bruno Berselli, Mar 31 2016

G.f.: (1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4)/(1 - x^5).
a(n) = a(n-5).
a(n) = 2^(n mod 5). - Bruno Berselli, Mar 31 2016

A008830 Discrete logarithm of n to the base 2 modulo 11.

Original entry on oeis.org

0, 1, 8, 2, 4, 9, 7, 3, 6, 5

Offset: 1

N. J. A. Sloane

Equivalently, a(n) is the multiplicative order of n with respect to base 2 (modulo 11), i.e., a(n) is the base-2 logarithm of the smallest k such that 2^k mod 11 = n. - Jon E. Schoenfield, Aug 21 2021

			From _Jon E. Schoenfield_, Aug 21 2021: (Start)
Sequence is a permutation of the 10 integers 0..9:
   k     2^k  2^k mod 11
  --  ------  ----------
   0       1           1  so a(1)  =  0
   1       2           2  so a(2)  =  1
   2       4           4  so a(4)  =  2
   3       8           8  so a(8)  =  3
   4      16           5  so a(5)  =  4
   5      32          10  so a(10) =  5
   6      64           9  so a(9)  =  6
   7     128           7  so a(7)  =  7
   8     256           3  so a(3)  =  8
   9     512           6  so a(6)  =  9
  10    1024           1
but a(1) = 0, so the sequence is finite with 10 terms.
(End)

I. M. Vinogradov, Elements of Number Theory, p. 220.

Eric Weisstein's World of Mathematics, Discrete Logarithm.

Cf. A036117.

j := 11; F := FiniteField(j); PrimitiveElement(F); [ Log(F!n) : n in [ 1..j-1 ]];

a:= n-> numtheory[mlog](n, 2, 11):
seq(a(n), n=1..10);  # Alois P. Heinz, Aug 21 2021

from sympy.ntheory import discrete_log
def a(n): return discrete_log(11, n, 2)
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Aug 13 2021

2^a(n) == n (mod 11). - Michael S. Branicky, Aug 13 2021

A202149 Triangle read by rows: T(n, k) = mod(2^k, n), where 1 <= k < n.

Original entry on oeis.org

0, 2, 1, 2, 0, 0, 2, 4, 3, 1, 2, 4, 2, 4, 2, 2, 4, 1, 2, 4, 1, 2, 4, 0, 0, 0, 0, 0, 2, 4, 8, 7, 5, 1, 2, 4, 2, 4, 8, 6, 2, 4, 8, 6, 2, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, 2, 4, 8, 2, 4, 8, 2, 4

Offset: 2

Alonso del Arte, Dec 12 2011

Rows indexed by odd primes end in 1 (and of course so do rows indexed by base 2 pseudoprimes, A001567). Of those rows, the ones that are permutations of the integers 1 to p - 1 correspond to primes with primitive root 2 (A001122).

			Triangle starts:
0
2 1
2 0 0
2 4 3 1
2 4 2 4  2
2 4 1 2  4 1
2 4 0 0  0 0 0
2 4 8 7  5 1 2 4
2 4 8 6  2 4 8 6 2
2 4 8 5 10 9 7 3 6 1
2 4 8 4  8 4 8 4 8 4 8

T. D. Noe, Rows n = 2..100, flattened

Cf. A036117, 2^n mod 11; A036118, 2^n mod 13; A201908, irregular triangle of 2^k mod (2n - 1).

ColumnForm[Table[PowerMod[2, k, n], {n, 2, 20}, {k, n - 1}], Center]

cp's OEIS Frontend

A201908 Irregular triangle of 2^k mod (2n-1).

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A010691 Period 2: repeat (1,10).

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A168429 a(n) = 4^n mod 11.

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A062116 a(n) = 2^n mod 17.

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A201912 Irregular triangle of 2^k mod prime(n).

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A187466 a(n) = 9^n mod 11.

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Maple

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A269266 a(n) = 2^n mod 31.

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