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 1-3 of 3 results.

A124223 Table T(n,k) = reciprocal of k modulo prime(n), for 1 <= k < prime(n), read by rows.

Original entry on oeis.org

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

Views

Author

Franklin T. Adams-Watters, Oct 20 2006

Keywords

Comments

T(n,k) = smallest m such that k*m == 1 (mod prime(n)); prime(n) is the n-th prime: A000040(n).

Examples

			From Alexander Elkins (alexander_elkins(AT)hotmail.com), Mar 26 2010: (Start)
Table begins:
  1;
  1,2;
  1,3,2,4;
  1,4,5,2,3,6;
  1,6,4,3,9,2,8,7,5,10;
  1,7,9,10,8,11,2,5,3,4,6,12;
  1,9,6,13,7,3,5,15,2,12,14,10,4,11,8,16;
  1,10,13,5,4,16,11,12,17,2,7,8,3,15,14,6,9,18;
  1,12,8,6,14,4,10,3,18,7,21,2,16,5,20,13,19,9,17,15,11,22;
  ... (End)

Links

Crossrefs

Cf. A124224, A102057, A000040, A006093 (row lengths).

Programs

  • Maple
    seq(seq(k^(ithprime(n)-2) mod ithprime(n), k=1..ithprime(n)-1), n=1..12); # Ridouane Oudra, Oct 04 2022
  • Mathematica
    Flatten[Table[PowerMod[n,-1,p],{p,Prime[Range[9]]},{n,p-1}]] (* Alexander Elkins (alexander_elkins(AT)hotmail.com), Mar 26 2010 *)
T[n_, k_] := ModularInverse[k, Prime[n]]; Table[T[n, k], {n, 1, 9}, {k, 1, Prime[n]-1}] // Flatten (* Jean-François Alcover, May 08 2017 *)
  • PARI
    row(n) = my(p=prime(n)); vector(p-1, k, lift(1/Mod(k, prime(n)))); \\ Michel Marcus, Feb 24 2023

Formula

From Alexander Elkins (alexander_elkins(AT)hotmail.com), Mar 26 2010: (Start)
T(n, 1) = 1;
T(n, T(n,k)) = k;
T(n, prime(n)-1) = prime(n)-1. (End)
T(n,k) = k^(prime(n)-2) mod prime(n), with 1 <= k < prime(n). - Ridouane Oudra, Oct 04 2022
From Ammar Khatab, Nov 07 2024: (Start)
T(n,2) = (prime(n)+1)/2;
T(n,3) = (2*prime(n)+1)/3 + 2*prime(n)/(sqrt(3)*3) * sin(4*(prime(n)+2)/3 * Pi);
T(n,8) = (8*prime(n)+1)/8 - prime(n)/8 * (prime(n) mod 8);
T(n,prime(n)-k) = prime(n) - T(n,k);
T(n,prime(n)-2) = (prime(n)-1)/2 ;
T(n,prime(n)-3) = (prime(n)-1)/3 - 2*prime(n)/(sqrt(3)*3) * sin(4*(prime(n)+2)/3 * Pi);
T(n,prime(n)-8) = -1/8 + prime(n)/8 * (prime(n) mod 8). (End)

A124224 Table T(n,k) = reciprocal of k-th number prime to n, modulo n, for 1 <= k <= phi(n).

Original entry on oeis.org

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

Views

Author

Franklin T. Adams-Watters, Oct 20 2006

Keywords

Comments

T(n,k) = smallest m such that A038566(n,k) * m = 1 (mod n).
For n>1 every row begins with 1 and ends with n-1. T(n,k) = A038566(n,k)^(phi(n) - 1) (mod n). - Geoffrey Critzer, Jan 03 2015

Examples

			The table T(n,k) starts:
n\k 1  2  2  3 4  5 6  7 8  9 10 11
1:  0
2:  1
3:  1  2
4:  1  3
5:  1  3  2  4
6:  1  5
7:  1  4  5  2 3  6
8:  1  3  5  7
9:  1  5  7  2 4  8
10: 1  7  3  9
11: 1  6  4  3 9  2 8  7 5 10
12: 1  5  7 11
13: 1  7  9 10 8 11 2  5 3  4  6 12
14: 1  5  3 11 9 13
15: 1  8  4 13 2 11 7 14
16: 1 11 13  7 9  3 5 15
...
n = 17: 1  9  6 13 7  3  5 15 2 12 14 10 4 11 8 16,
n = 18: 1 11 13  5 7 17,
n = 19: 1 10 13  5 4 16 11 12 17 2 7 8 3 15 14 6 9 18,
n = 20: 1 7 3 9 11 17 13 19.
... reformatted (extended and corrected), - _Wolfdieter Lang_, Oct 06 2016

Links

Crossrefs

Cf. A124223, A102057, A038566, A000010 (row lengths), A023896 (row sums after first)

Programs

  • Maple
    0,seq(seq(i^(-1) mod m, i = select(t->igcd(t,m)=1, [$1..m-1])),m=1..100); # Robert Israel, May 18 2014
  • Mathematica
    Table[nn = n; a = Select[Range[nn], CoprimeQ[#, nn] &];
PowerMod[a, -1, nn], {n, 1, 20}] // Grid (* Geoffrey Critzer, Jan 03 2015 *)

Formula

T(n,k) * A038566(n,k) = 1 (mod n), for n >=1 and k=1..A000010(n). - Wolfdieter Lang, Oct 06 2016

A289251 Triangle T(n, k), n > 0 and 0 <= k < n, read by rows; if gcd(n, k) = 1, then T(n, k) = modular inverse of k (mod n), otherwise T(n, k) = k.

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, Jun 29 2017

Keywords

Comments

The n-th row has n terms, and is a self-inverse permutation of the first n nonnegative numbers.
T(n, 0) = 0 for any n > 0.
T(n, 1) = 1 for any n > 1.
T(n, n-1) = n-1 for any n > 0.
If n > 0 and gcd(n, k) = 1 then T(n, k) = A102057(n, k).
T(prime(n), k) = A124223(n, k) for any n > 0 and k in 1..prime(n)-1.

Examples

			The first rows are:
n\k  0 1 2 3 4 5 6 7 8 9
1    0
2    0 1
3    0 1 2
4    0 1 2 3
5    0 1 3 2 4
6    0 1 2 3 4 5
7    0 1 4 5 2 3 6
8    0 1 2 3 4 5 6 7
9    0 1 5 3 7 2 6 4 8
10   0 1 2 7 4 5 6 3 8 9

Links

Crossrefs

Cf. A102057, A124223.

Programs

  • Mathematica
    T[n_, k_] := If[GCD[n, k] == 1, PowerMod[k, -1, n], k];
Table[T[n, k], {n, 1, 13}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Oct 31 2017 *)
  • PARI
    T(n, k) = if (gcd(n, k)==1, lift(1/Mod(k, n)), k)
Showing 1-3 of 3 results.

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