A345873 -id:A345873 - cp's OEIS frontend

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

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

N. J. A. Sloane, Jun 19 2021

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.

			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]
...

Cf. A003989, A050873, A345416, A345417, A345418.

Cf. also A345872, A345873.

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;

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 *)

A345872 Lexicographically earliest sequence of positive integers such that the values A345415(a(n), a(n+1)) are all distinct.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 5, 8, 11, 13, 11, 18, 23, 13, 15, 13, 21, 19, 21, 23, 18, 31, 19, 30, 37, 30, 41, 39, 23, 25, 34, 43, 28, 43, 41, 43, 45, 37, 35, 37, 50, 47, 38, 41, 54, 61, 52, 61, 59, 57, 55, 53, 64, 53, 70, 53, 76, 37, 69, 49, 66, 73, 71, 69, 71, 73, 75

Offset: 1

Rémy Sigrist, Jun 27 2021

nonn

When writing gcd(a(n), a(n+1)) as u*a(n) + v*a(n+1) where u, v are minimal, the u's are all distinct.

			The table A345415(n, k) begins:
  n\k|  1  2   3   4
  ---+--------------
    1|  0  1   1   1
    2|  0  0  -1   1
    2|  0  1   0  -1
    2|  0  0   1   0
For n = 1:
- we can choose a(1) = 1.
For n = 2:
- we can choose a(2) = 1,
- A345415(a(1), a(2)) = 0.
For n = 3:
- a(3) must be different from 1,
- we can choose a(3) = 2,
- A345415(a(2), a(3)) = 1.
For n = 4:
- a(4) must be different from 1 and from 2,
- we can choose a(4) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A345872

Cf. A345415, A345873.

PARI
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cp's OEIS Frontend

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

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A345872 Lexicographically earliest sequence of positive integers such that the values A345415(a(n), a(n+1)) are all distinct.

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cp's OEIS Frontend

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.

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Author

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Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A345872 Lexicographically earliest sequence of positive integers such that the values A345415(a(n), a(n+1)) are all distinct.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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