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.

A345415 Table read by upward antidiagonals: Given m, n >= 1, write gcd(m,n) as d = u*m+v*n where u, v are minimal; T(m,n) = u.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, -1, 1, 0, 0, 0, 1, 1, 0, 1, 1, -1, -2, 1, 0, 0, -1, 0, 2, 1, 1, 0, 1, 0, 1, -1, 1, -3, 1, 0, 0, 1, 1, 0, -1, -2, 1, 1, 0, 1, -1, -1, 1, -1, 2, 3, -4, 1, 0, 0, 0, 0, -2, 0, 3, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, -1, -3, -2, -3, -5, 1, 0, 0, -1, 1, -1, 1, 0, -1, 2, -2, 4, 1, 1
Offset: 1

Views

Author

N. J. A. Sloane, Jun 19 2021

Keywords

Comments

The gcd is given in A003989, and v is given in A345416. Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when m=n. If we ignore the diagonal, the v table is the transpose of the u table.

Examples

			The gcd table (A003989) begins:
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2]
[1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1]
[1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4]
[1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1]
[1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2]
[1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1]
[1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8]
[1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1]
[1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1]
[1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4]
...
The u table (this entry) begins:
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[0, 0, -1, 1, -2, 1, -3, 1, -4, 1, -5, 1, -6, 1, -7, 1]
[0, 1, 0, -1, 2, 1, -2, 3, 1, -3, 4, 1, -4, 5, 1, -5]
[0, 0, 1, 0, -1, -1, 2, 1, -2, -2, 3, 1, -3, -3, 4, 1]
[0, 1, -1, 1, 0, -1, 3, -3, 2, 1, -2, 5, -5, 3, 1, -3]
[0, 0, 0, 1, 1, 0, -1, -1, -1, 2, 2, 1, -2, -2, -2, 3]
[0, 1, 1, -1, -2, 1, 0, -1, 4, 3, -3, -5, 2, 1, -2, 7]
[0, 0, -1, 0, 2, 1, 1, 0, -1, -1, -4, -1, 5, 2, 2, 1]
[0, 1, 0, 1, -1, 1, -3, 1, 0, -1, 5, -1, 3, -3, 2, -7]
[0, 0, 1, 1, 0, -1, -2, 1, 1, 0, -1, -1, 4, 3, -1, -3]
[0, 1, -1, -1, 1, -1, 2, 3, -4, 1, 0, -1, 6, -5, -4, 3]
[0, 0, 0, 0, -2, 0, 3, 1, 1, 1, 1, 0, -1, -1, -1, -1]
...
The v table (A345416) begins:
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
[1, -1, 1, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1]
[1, 1, -1, 1, 1, 1, -1, 0, 1, 1, -1, 0, 1, 1, -1, 0]
[1, -2, 2, -1, 1, 1, -2, 2, -1, 0, 1, -2, 2, -1, 0, 1]
[1, 1, 1, -1, -1, 1, 1, 1, 1, -1, -1, 0, 1, 1, 1, -1]
[1, -3, -2, 2, 3, -1, 1, 1, -3, -2, 2, 3, -1, 0, 1, -3]
[1, 1, 3, 1, -3, -1, -1, 1, 1, 1, 3, 1, -3, -1, -1, 0]
[1, -4, 1, -2, 2, -1, 4, -1, 1, 1, -4, 1, -2, 2, -1, 4]
[1, 1, -3, -2, 1, 2, 3, -1, -1, 1, 1, 1, -3, -2, 1, 2]
[1, -5, 4, 3, -2, 2, -3, -4, 5, -1, 1, 1, -5, 4, 3, -2]
[1, 1, 1, 1, 5, 1, -5, -1, -1, -1, -1, 1, 1, 1, 1, 1]
...

Crossrefs

Cf. A003989, A050873, A345416, A345417, A345418.
Cf. also A345872, A345873.

Programs

  • Maple
    mygcd:=proc(a,b) local d,s,t; d := igcdex(a,b,`s`,`t`); [a,b,d,s,t]; end;
gcd_rowu:=(m,M)->[seq(mygcd(m,n)[4],n=1..M)];
for m from 1 to 12 do lprint(gcd_rowu(m,16)); od;
  • Mathematica
    T[m_, n_] := Module[{u, v}, MinimalBy[{u, v} /. Solve[u^2 + v^2 <= 26 && u*m + v*n == GCD[m, n], {u, v}, Integers], #.#&][[1, 1]]];
Table[T[m - n + 1, n], {m, 1, 13}, {n, 1, m}] // Flatten (* Jean-François Alcover, Mar 27 2023 *)

A345417 Table read by upward antidiagonals: Given m, n >= 1, write gcd(prime(m),prime(n)) as d = u*prime(m)+v*prime(n) where u, v are minimal; T(m,n) = u.

Original entry on oeis.org

0, 1, -1, 1, 0, -2, 1, -1, 2, -3, 1, 1, 0, -2, -5, 1, -1, -2, 3, 4, -6, 1, 1, 1, 0, -2, -4, -8, 1, -1, 2, 2, -3, -5, 6, -9, 1, 1, -2, -1, 0, 2, 7, -6, -11, 1, -1, -1, -2, -5, 6, 5, 4, 8, -14, 1, -1, 2, 3, 2, 0, -3, -8, -9, 10, -15, 1, 1, -1, -3, -4, -3, 4, 7, 10, 6, -10, -18
Offset: 1

Views

Author

N. J. A. Sloane, Jun 19 2021

Keywords

Comments

The gcd is 1 unless m=n when it is m; v is given in A345418. Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when m=n. If we ignore the diagonal, the v table is the transpose of the u table.

Examples

			The u table (this entry) begins:
[0, -1, -2, -3, -5, -6, -8, -9, -11, -14, -15, -18, -20, -21, -23, -26]
[1, 0, 2, -2, 4, -4, 6, -6, 8, 10, -10, -12, 14, -14, 16, 18]
[1, -1, 0, 3, -2, -5, 7, 4, -9, 6, -6, 15, -8, -17, 19, -21]
[1, 1, -2, 0, -3, 2, 5, -8, 10, -4, 9, 16, 6, -6, -20, -15]
[1, -1, 1, 2, 0, 6, -3, 7, -2, 8, -14, -10, 15, 4, -17, -24]
[1, 1, 2, -1, -5, 0, 4, 3, -7, 9, 12, -17, 19, 10, -18, -4]
[1, -1, -2, -2, 2, -3, 0, 9, -4, 12, 11, -13, -12, -5, -11, 25]
[1, 1, -1, 3, -4, -2, -8, 0, -6, -3, -13, 2, 13, -9, 5, 14]
[1, -1, 2, -3, 1, 4, 3, 5, 0, -5, -4, -8, -16, 15, -2, -23]
[1, -1, -1, 1, -3, -4, -7, 2, 4, 0, 15, -14, 17, 3, 13, 11]
[1, 1, 1, -2, 5, -5, -6, 8, 3, -14, 0, 6, 4, -18, -3, 12]
[1, 1, -2, -3, 3, 6, 6, -1, 5, 11, -5, 0, 10, 7, 14, -10]
...
The v table (A345418) begins:
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[-1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, -1]
[-2, 2, 1, -2, 1, 2, -2, -1, 2, -1, 1, -2, 1, 2, -2, 2]
[-3, -2, 3, 1, 2, -1, -2, 3, -3, 1, -2, -3, -1, 1, 3, 2]
[-5, 4, -2, -3, 1, -5, 2, -4, 1, -3, 5, 3, -4, -1, 4, 5]
[-6, -4, -5, 2, 6, 1, -3, -2, 4, -4, -5, 6, -6, -3, 5, 1]
[-8, 6, 7, 5, -3, 4, 1, -8, 3, -7, -6, 6, 5, 2, 4, -8]
[-9, -6, 4, -8, 7, 3, 9, 1, 5, 2, 8, -1, -6, 4, -2, -5]
[-11, 8, -9, 10, -2, -7, -4, -6, 1, 4, 3, 5, 9, -8, 1, 10]
[-14, 10, 6, -4, 8, 9, 12, -3, -5, 1, -14, 11, -12, -2, -8, -6]
[-15, -10, -6, 9, -14, 12, 11, -13, -4, 15, 1, -5, -3, 13, 2, -7]
[-18, -12, 15, 16, -10, -17, -13, 2, -8, -14, 6, 1, -9, -6, -11, 7]
...

Crossrefs

Cf. A003989, A050873, A345415, A345416, A345418, A345419-A345422.

A345416 Table read by upward antidiagonals: Given m, n >= 1, write gcd(m,n) as d = u*m+v*n where u, v are minimal; T(m,n) = v.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, -1, 1, 0, 1, 1, 1, 0, 0, 1, -2, -1, 1, 1, 0, 1, 1, 2, 1, -1, 0, 0, 1, -3, 1, -1, 1, 0, 1, 0, 1, 1, -2, -1, 1, 1, 1, 0, 0, 1, -4, 3, 2, -1, 1, -1, -1, 1, 0, 1, 1, 1, 1, 3, 1, -2, 0, 0, 0, 0, 1, -5, -3, -2, -3, -1, 1, 2, 1, 1, 1, 0, 1, 1, 4, -2, 2, -1, 1, 1, -1, 1, -1, 0, 0
Offset: 1

Views

Author

N. J. A. Sloane, Jun 19 2021

Keywords

Comments

The gcd is given in A003989, and u is given in A345415. Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when m=n. If we ignore the diagonal, the v table is the transpose of the u table.

Examples

			The gcd table (A003989) begins:
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2]
[1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1]
[1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4]
[1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1]
[1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2]
[1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1]
[1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8]
[1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1]
[1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1]
[1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4]
...
The u table (A345415) begins:
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[0, 0, -1, 1, -2, 1, -3, 1, -4, 1, -5, 1, -6, 1, -7, 1]
[0, 1, 0, -1, 2, 1, -2, 3, 1, -3, 4, 1, -4, 5, 1, -5]
[0, 0, 1, 0, -1, -1, 2, 1, -2, -2, 3, 1, -3, -3, 4, 1]
[0, 1, -1, 1, 0, -1, 3, -3, 2, 1, -2, 5, -5, 3, 1, -3]
[0, 0, 0, 1, 1, 0, -1, -1, -1, 2, 2, 1, -2, -2, -2, 3]
[0, 1, 1, -1, -2, 1, 0, -1, 4, 3, -3, -5, 2, 1, -2, 7]
[0, 0, -1, 0, 2, 1, 1, 0, -1, -1, -4, -1, 5, 2, 2, 1]
[0, 1, 0, 1, -1, 1, -3, 1, 0, -1, 5, -1, 3, -3, 2, -7]
[0, 0, 1, 1, 0, -1, -2, 1, 1, 0, -1, -1, 4, 3, -1, -3]
[0, 1, -1, -1, 1, -1, 2, 3, -4, 1, 0, -1, 6, -5, -4, 3]
[0, 0, 0, 0, -2, 0, 3, 1, 1, 1, 1, 0, -1, -1, -1, -1]
...
The v table (this entry) begins:
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
[1, -1, 1, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1]
[1, 1, -1, 1, 1, 1, -1, 0, 1, 1, -1, 0, 1, 1, -1, 0]
[1, -2, 2, -1, 1, 1, -2, 2, -1, 0, 1, -2, 2, -1, 0, 1]
[1, 1, 1, -1, -1, 1, 1, 1, 1, -1, -1, 0, 1, 1, 1, -1]
[1, -3, -2, 2, 3, -1, 1, 1, -3, -2, 2, 3, -1, 0, 1, -3]
[1, 1, 3, 1, -3, -1, -1, 1, 1, 1, 3, 1, -3, -1, -1, 0]
[1, -4, 1, -2, 2, -1, 4, -1, 1, 1, -4, 1, -2, 2, -1, 4]
[1, 1, -3, -2, 1, 2, 3, -1, -1, 1, 1, 1, -3, -2, 1, 2]
[1, -5, 4, 3, -2, 2, -3, -4, 5, -1, 1, 1, -5, 4, 3, -2]
[1, 1, 1, 1, 5, 1, -5, -1, -1, -1, -1, 1, 1, 1, 1, 1]
...

Crossrefs

Cf. A003989, A050873, A345415, A345417, A345418.

Programs

  • Maple
    mygcd:=proc(a,b) local d,s,t; d := igcdex(a,b,`s`,`t`); [a,b,d,s,t]; end;
gcd_rowv:=(m,M)->[seq(mygcd(m,n)[5],n=1..M)];
for m from 1 to 12 do lprint(gcd_rowv(m,16)); od;
  • Mathematica
    T[m_, n_] := Module[{u, v}, MinimalBy[{u, v} /. Solve[u^2 + v^2 <= 26 && u*m + v*n == GCD[m, n], {u, v}, Integers], #.#&][[1, 2]]];
Table[T[m - n + 1, n], {m, 1, 13}, {n, 1, m}] // Flatten (* Jean-François Alcover, Mar 27 2023 *)
Showing 1-3 of 3 results.

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