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.

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

Original entry on oeis.org

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

Views

Author

T. D. Noe, Dec 07 2011

Keywords

Comments

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.

Examples

			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.

References

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

Links

Crossrefs

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).

Programs

  • GAP
    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
  • Mathematica
    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}]]

Formula

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

A201911 Irregular triangle of 7^k mod prime(n).

Original entry on oeis.org

1, 1, 1, 2, 4, 3, 0, 1, 7, 5, 2, 3, 10, 4, 6, 9, 8, 1, 7, 10, 5, 9, 11, 12, 6, 3, 8, 4, 2, 1, 7, 15, 3, 4, 11, 9, 12, 16, 10, 2, 14, 13, 6, 8, 5, 1, 7, 11, 1, 7, 3, 21, 9, 17, 4, 5, 12, 15, 13, 22, 16, 20, 2, 14, 6, 19, 18, 11, 8, 10, 1, 7, 20, 24, 23, 16, 25
Offset: 1

Views

Author

T. D. Noe, Dec 07 2011

Keywords

Comments

Except for the fourth row, the first term of each row is 1. Many sequences are in this one: starting at A036132 (mod 71) and A070404 (mod 11).

Examples

			The first 9 rows are:
  1
  1
  1, 2,  4,  3
  0
  1, 7,  5,  2, 3, 10,  4,  6,  9,  8
  1, 7, 10,  5, 9, 11, 12,  6,  3,  8,  4,  2
  1, 7, 15,  3, 4, 11,  9, 12, 16, 10,  2, 14, 13,  6, 8,  5
  1, 7, 11
  1, 7,  3, 21, 9, 17,  4,  5, 12, 15, 13, 22, 16, 20, 2, 14, 6, 19, 18, 11, 8, 10

Links

Crossrefs

Cf. A201908 (2^k), A201909 (3^k), A201910 (5^k).
Cf. A070404 (11), A070405 (13), A070407 (17), A070409 (23), A070413 (29), A070415 (31), A070420 (37), A070422 (39), A070424 (41), A070425 (43), A070429 (47), A036132 (71).

Programs

  • GAP
    P:=Filtered([1..350],IsPrime);;
R:=List([1..Length(P)],n->OrderMod(7,P[n]));;
Flat(Concatenation([1,1,1,2,4,3,0],List([5..10],n->List([0..R[n]-1],k->PowerMod(7,k,P[n]))))); # Muniru A Asiru, Feb 01 2019
  • Mathematica
    nn = 10; p = 7; 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]]}]]

A201909 Irregular triangle of 3^k mod prime(n).

Original entry on oeis.org

1, 0, 1, 3, 4, 2, 1, 3, 2, 6, 4, 5, 1, 3, 9, 5, 4, 1, 3, 9, 1, 3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6, 1, 3, 9, 8, 5, 15, 7, 2, 6, 18, 16, 10, 11, 14, 4, 12, 17, 13, 1, 3, 9, 4, 12, 13, 16, 2, 6, 18, 8, 1, 3, 9, 27, 23, 11, 4, 12, 7, 21, 5, 15
Offset: 1

Views

Author

T. D. Noe, Dec 07 2011

Keywords

Comments

The row lengths are in A062117. Except for the second row, the first term of each row is 1. Many sequences are in this one: starting at A036119 (mod 17) and A070341 (mod 11).

Examples

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

Links

Crossrefs

Cf. A062117, A201908 (2^k), A201910 (5^k), A201911 (7^k).
Cf. A070352 (5), A033940 (7), A070341 (11), A168399 (13), A036119 (17), A070342 (19), A070356 (23), A070344 (29), A036123 (31), A070346 (37), A070361 (41), A036126 (43), A070364 (47), A036134 (79), A036136 (89), A036142 (113), A036143 (127), A036145 (137), A036158 (199), A036160 (223).

Programs

  • GAP
    P:=Filtered([1..350],IsPrime);;
R:=List([1..Length(P)],n->OrderMod(7,P[n]));;
Flat(Concatenation([1,1,1,2,4,3,0],List([5..10],n->List([0..R[n]-1],k->PowerMod(7,k,P[n]))))); # Muniru A Asiru, Feb 01 2019
  • Mathematica
    nn = 10; p = 3; 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]]}]]

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

Views

Author

T. D. Noe, Dec 17 2011

Keywords

Comments

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.

Examples

			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;

Links

Crossrefs

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).

Programs

  • GAP
    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
  • Mathematica
    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]]}]]

A323873 Irregular triangle of 11^k mod prime(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 2, 0, 1, 11, 4, 5, 3, 7, 12, 2, 9, 8, 10, 6, 1, 11, 2, 5, 4, 10, 8, 3, 16, 6, 15, 12, 13, 7, 9, 14, 1, 11, 7, 1, 11, 6, 20, 13, 5, 9, 7, 8, 19, 2, 22, 12, 17, 3, 10, 18, 14, 16, 15, 4, 21, 1, 11, 5, 26, 25, 14, 9, 12, 16, 2, 22, 10, 23, 21, 28
Offset: 1

Views

Author

Muniru A Asiru, Feb 04 2019

Keywords

Comments

Length of the n-th row (n != 5) is the order of 11 modulo the n-th prime.
Except for the fifth row, the first term of each row is 1.

Examples

			The first 9 rows are:
  1;
  1,  2;
  1;
  1,  4, 2;
  0;
  1, 11, 4,  5,  3,  7, 12, 2,  9,  8, 10,  6;
  1, 11, 2,  5,  4, 10,  8, 3, 16,  6, 15, 12, 13,  7, 9, 14;
  1, 11, 7;
  1, 11, 6, 20, 13,  5,  9, 7,  8, 19,  2, 22, 12, 17, 3, 10, 18, 14, 16, 15, 4, 21;
  ...

Links

Crossrefs

Cf. A201908 (2^k), A201909 (3^k), A201910 (5^k), A201911 (7^k), this sequence (11^k), A323874 (13^k).
Cf. A000040.

Programs

  • GAP
    A000040:=Filtered([1..350],IsPrime);; p:=5;;
R:=List([1..Length(A000040)],n->OrderMod(A000040[p],A000040[n]));;
a1:=List([1..p-1],n->List([0..R[n]-1],k->PowerMod(A000040[p],k,A000040[n])));;
a:=Flat(Concatenation(a1,[0],List([p+1..2*p],n->List([0..R[n]-1],k->PowerMod(A000040[p],k,A000040[n])))));; Print(a);
  • Maple
    T:= n-> (p-> `if`(p=11, 0, seq(11&^k mod p,
         k=0..numtheory[order](11, p)-1)))(ithprime(n)):
seq(T(n), n=1..15);  # Alois P. Heinz, Feb 06 2019
  • Mathematica
    Table[If[p == 11, {0}, Array[PowerMod[11, #, p] &, MultiplicativeOrder[11, p], 0]], {p, Prime@ Range@ 10}] (* Michael De Vlieger, Feb 25 2019 *)

A323874 Irregular triangle of 13^k mod prime(n).

Original entry on oeis.org

1, 1, 1, 3, 4, 2, 1, 6, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 0, 1, 13, 16, 4, 1, 13, 17, 12, 4, 14, 11, 10, 16, 18, 6, 2, 7, 15, 5, 8, 9, 3, 1, 13, 8, 12, 18, 4, 6, 9, 2, 3, 16, 1, 13, 24, 22, 25, 6, 20, 28, 16, 5, 7, 4, 23, 9, 1, 13, 14, 27, 10, 6, 16, 22, 7, 29, 5
Offset: 1

Views

Author

Muniru A Asiru, Feb 04 2019

Keywords

Comments

Length of the n-th row (n != 6) is the order of 13 modulo the n-th prime.
Except for the sixth row, the first term of each row is 1.

Examples

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

Crossrefs

Cf. A000040.
Cf. A201908 (2^k), A201909 (3^k), A201910 (5^k), A201911 (7^k), A323873 (11^k), this sequence (13^k).

Programs

  • GAP
    A000040:=Filtered([1..350],IsPrime);; p:=6;;
R:=List([1..Length(A000040)],n->OrderMod(A000040[p],A000040[n]));;
a1:=List([1..p-1],n->List([0..R[n]-1],k->PowerMod(A000040[p],k,A000040[n])));;
a:=Flat(Concatenation(a1,[0],List([p+1..2*p],n->List([0..R[n]-1],k->PowerMod(A000040[p],k,A000040[n])))));; Print(a);
  • Maple
    T:= n-> (p-> `if`(p=13, 0, seq(13&^k mod p,
         k=0..numtheory[order](13, p)-1)))(ithprime(n)):
seq(T(n), n=1..15);  # Alois P. Heinz, Feb 06 2019
  • Mathematica
    With[{q = 13}, Table[If[p == q, {0}, Array[PowerMod[q, #, p] &, MultiplicativeOrder[q, p], 0]], {p, Prime@ Range@ 11}]] // Flatten (* Michael De Vlieger, Feb 25 2019 *)
Showing 1-6 of 6 results.

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