A345415 -id:A345415 - cp's OEIS frontend

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

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
```
See Links section.
```

A345873 a(n) = A345415(A345872(n), A345872(n+1)).

Original entry on oeis.org

0, 1, -1, 2, 3, -2, -3, -4, 6, -5, 5, 9, 4, 7, -6, -8, -9, 10, 11, -7, -12, 8, -11, -16, 13, -15, -19, -10, 12, 15, 19, -13, 20, -20, 21, 22, 14, -17, 18, 23, 16, 17, -14, -25, 26, -23, 27, -29, -28, -27, -26, 29, -24, -33, 25, 33, -18, 28, -22, 31, -21, -35

Offset: 1

Rémy Sigrist, Jun 27 2021

sign

All terms are distinct.

			a(3) = A345415(A345872(3), A345872(4)) = A345415(2, 3) = -1.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A345873

Cf. A345415, A345872.

PARI
```
See Links section.
```

A345417 Table read by upward antidiagonals: Given m, n >= 1, write gcd(prime(m),prime(n)) as d = uprime(m)+vprime(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

N. J. A. Sloane, Jun 19 2021

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.

			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]
...

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

A345428 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = ux+vy with u,v minimal; a(n) = sum of the values of u+v.

Original entry on oeis.org

1, 4, 7, 12, 15, 22, 23, 32, 33, 38, 41, 54, 41, 54, 55, 60, 65, 64, 47, 70, 53, 60, 69, 102, 47, 36, 35, 22, 41, 70, 47, 80, 13, -4, 15, -8, -49, -22, -49, -46, -53, -36, -141, -32, -57, -76, -63, -66, -205, -298, -275, -252, -289, -298

Offset: 1

N. J. A. Sloane, Jun 22 2021

sign

Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when x=y.

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A345415-A345427.

T:= proc(x,y) option remember; local g,u0,v0,t0,t1,t2;
   g:= igcd(x,y);
   if g > 1 then return procname(x/g,y/g) fi;
   v0:= y^(-1) mod x;
   u0:= (1-y*v0)/x;
   t0:= (v0*x-u0*y)/(x^2+y^2);
   t1:= floor(t0);
   if t0 < t1 + 1/2 then u0+v0 + t1*(y-x)
   else u0+v0 + (t1+1)*(y-x)
   fi
end proc:
R:= 1: v:= 1:
for n from 2 to 100 do v:= v+1+2*add(T(i,n),i=1..n-1); R:= R,v od:
R; # Robert Israel, Mar 28 2023

T[x_, y_] := T[x, y] = Module[{u, v}, MinimalBy[{u, v} /. Solve[u^2 + v^2 <= x^2 + y^2 && u*x + v*y == GCD[x, y], {u, v}, Integers], #.# &]];
a[n_] := a[n] = Sum[T[x, y][[1]]//Total, {x, 1, n}, {y, 1, n}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 54}] (* Jean-François Alcover, Mar 28 2023 *)

from sympy.core.numbers import igcdex
def A345428(n): return sum(u+v for u, v, w in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1))) # Chai Wah Wu, Jun 24 2021

A345434 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = ux+vy with u,v minimal; a(n) = sum of the values of u^2+v^2.

Original entry on oeis.org

1, 4, 11, 20, 47, 62, 135, 196, 313, 394, 685, 838, 1317, 1578, 1991, 2484, 3573, 4084, 5595, 6410, 7621, 8792, 11505, 12710, 15539, 17536, 20619, 23018, 28417, 30650, 37215, 41308, 46405, 51072, 57607, 61596, 72927, 79670, 88055, 94618, 109799, 116312, 134067, 143952, 155287

Offset: 1

N. J. A. Sloane, Jun 22 2021

nonn

Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when x=y.

Cf. A345415-A345433.

from sympy.core.numbers import igcdex
def A345434(n): return sum(u**2+v**2 for u, v, w in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1))) # Chai Wah Wu, Jun 22 2021

A345416 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) = 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

N. J. A. Sloane, Jun 19 2021

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.

			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]
...

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

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;

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 *)

A345427 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = ux+vy with u,v minimal; a(n) = sum of the values of v.

Original entry on oeis.org

1, 3, 5, 8, 10, 14, 15, 20, 21, 24, 26, 33, 27, 34, 35, 38, 41, 41, 33, 45, 37, 41, 46, 63, 36, 31, 31, 25, 35, 50, 39, 56, 23, 15, 25, 14, -6, 8, -5, -3, -6, 3, -49, 6, -6, -15, -8, -9, -78, -124, -112, -100, -118, -122, -133, -109, -110, -139, -127, -117, -237, -166, -185, -218, -171, -215

Offset: 1

N. J. A. Sloane, Jun 22 2021

sign

Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when x=y.

Cf. A345415-A345428.

T[x_, y_] := T[x, y] = Module[{u, v}, MinimalBy[{u, v} /. Solve[u^2 + v^2 <= x^2 + y^2 && u*x + v*y == GCD[x, y], {u, v}, Integers], #.# &]];
a[n_] := a[n] = Sum[T[x, y][[1, 2]], {x, 1, n}, {y, 1, n}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 62}] (* Jean-François Alcover, Mar 28 2023 *)

from sympy.core.numbers import igcdex
def A345427(n): return sum(v for u, v, w in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1))) # Chai Wah Wu, Jun 22 2021

A345433 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = ux+vy with u,v minimal; a(n) = sum of the values of |v|.

Original entry on oeis.org

1, 3, 7, 12, 22, 30, 49, 66, 91, 114, 160, 189, 253, 300, 355, 420, 529, 597, 733, 819, 931, 1053, 1252, 1365, 1564, 1737, 1945, 2121, 2437, 2604, 2965, 3222, 3511, 3813, 4147, 4398, 4912, 5292, 5701, 6039, 6670, 7011, 7705, 8160, 8638, 9201, 10030, 10479, 11272, 11856, 12568, 13212, 14266

Offset: 1

N. J. A. Sloane, Jun 22 2021

nonn

Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when x=y.

Cf. A345415-A345434.

from sympy.core.numbers import igcdex
def A345433(n): return sum(abs(v) for u, v, w in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1))) # Chai Wah Wu, Jun 22 2021

A345418 Table read by upward antidiagonals: Given m, n >= 1, write gcd(prime(m),prime(n)) as d = uprime(m)+vprime(n) where u, v are minimal; T(m,n) = v.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 19 2021

The gcd is 1 unless m=n when it is m; u is given in A345417. 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 u table (A345417) 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 (this entry) 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]
...

Cf. A003989, A050873, A345415, A345416, A345417, A345419, A345420, A345421, A345422.

A345423 For 1<=x<=n, 1<=y<=n, with gcd(x,y)=1, write 1 = gcd(x,y) = ux+vy with u,v minimal; a(n) = sum of the values of u.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 5, 7, 6, 6, 7, 9, 2, 7, 5, 3, 5, 2, -7, 1, -9, -8, -4, 4, -25, -25, -26, -40, -31, -19, -31, -17, -53, -65, -57, -71, -92, -71, -79, -91, -95, -85, -138, -88, -100, -115, -109, -125, -195, -215, -207, -191, -210, -213, -227, -199, -193, -233, -222, -238

Offset: 1

N. J. A. Sloane, Jun 22 2021

sign

Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when x=y.

Cf. A345415-A345428.

mygcd:=proc(a,b) local d,s,t; d := igcdex(a,b,`s`,`t`); [a,b,d,s,t]; end;
ansu:=[]; ansv:=[]; ansb:=[];
for N from 1 to 80 do
tu:=0; tv:=0; tb:=0;
for x from 1 to N do
for y from 1 to N do
if igcd(x,y)=1 then
   tu:=tu + mygcd(x,y)[4];
   tv:=tv + mygcd(x,y)[5];
   tb:=tb + mygcd(x,y)[4] + mygcd(x,y)[5];
fi;
od: od:
ansu:=[op(ansu),tu];
ansv:=[op(ansv),tv];
ansb:=[op(ansb),tb];
od:
ansu; # the present sequence
ansv; # A345424
ansb; # A345425
# for A345426, A345427, A345428, omit the "igcd(x,y)=1" test

T[x_, y_] := T[x, y] = Module[{u, v}, MinimalBy[{u, v} /. Solve[u^2 + v^2 <= x^2 + y^2 && u*x + v*y == 1, {u, v}, Integers], #.# &]];
a[n_] := a[n] = Sum[If[GCD[x, y] == 1, T[x, y][[1, 1]], 0], {x, 1, n}, {y, 1, n}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 28 2023 *)

from sympy.core.numbers import igcdex
def A345423(n): return sum(u for u, v, w in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1)) if w == 1) # Chai Wah Wu, Aug 21 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A345873 a(n) = A345415(A345872(n), A345872(n+1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

A345428 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = u*x+v*y with u,v minimal; a(n) = sum of the values of u+v.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

A345434 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = u*x+v*y with u,v minimal; a(n) = sum of the values of u^2+v^2.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A345427 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = u*x+v*y with u,v minimal; a(n) = sum of the values of v.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

A345433 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = u*x+v*y with u,v minimal; a(n) = sum of the values of |v|.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

A345418 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) = v.

Views

Author

Keywords

Comments

Examples

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

A345428 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = ux+vy with u,v minimal; a(n) = sum of the values of u+v.

A345434 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = ux+vy with u,v minimal; a(n) = sum of the values of u^2+v^2.

A345416 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) = v.

A345427 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = ux+vy with u,v minimal; a(n) = sum of the values of v.

A345433 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = ux+vy with u,v minimal; a(n) = sum of the values of |v|.

A345418 Table read by upward antidiagonals: Given m, n >= 1, write gcd(prime(m),prime(n)) as d = uprime(m)+vprime(n) where u, v are minimal; T(m,n) = v.

A345423 For 1<=x<=n, 1<=y<=n, with gcd(x,y)=1, write 1 = gcd(x,y) = ux+vy with u,v minimal; a(n) = sum of the values of u.