A347684 -id:A347684 - cp's OEIS frontend

A347687 Triangle read by rows: T(n,k) (1<=k<=n) = (r(n+k)-r(k))/(2*n), where {r(i), i>=1} is the n-th row of array A347684.

0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 2, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 3, 2, 2, 3, 1, 0, 1, 0, 3, 0, 3, 0, 1, 0, 1, 4, 0, 2, 2, 0, 4, 1, 0, 1, 0, 3, 0, 0, 0, 3, 0, 1, 0, 1, 5, 4, 3, 2, 2, 3, 4, 5, 1, 0, 1, 0, 0, 0, 5, 0, 5, 0, 0, 0, 1, 0, 1, 6, 4, 3, 5, 2, 2, 5, 3, 4, 6, 1, 0

Offset: 1

N. J. A. Sloane, Sep 19 2021

Stated another way, T(n,k) = (1/(2*n)) * (A347684(n,k+n) - A347684(n,k)) for 1 <= k <= n.
Conjecture: Row n is not just the rescaled differences between the first two blocks of n terms of row n of A347684, but is in fact the rescaled differences between any two successive blocks of n terms that start at positions == 1 (mod n). See also the comments in A347684.
It would be nice to have an independent characterization of this triangle. There is a lot of structure - see the link for the first 40 rows.
Except for the final 0 in each row, the rows begin with 1 and are palindromic. T(n,k) = 0 iff gcd(n,k) > 1.

			The initial rows are:
1,  [0],
2,  [1, 0],
3,  [1, 1, 0],
4,  [1, 0, 1, 0],
5,  [1, 2, 2, 1, 0],
6,  [1, 0, 0, 0, 1, 0],
7,  [1, 3, 2, 2, 3, 1, 0],
8,  [1, 0, 3, 0, 3, 0, 1, 0],
9,  [1, 4, 0, 2, 2, 0, 4, 1, 0],
10, [1, 0, 3, 0, 0, 0, 3, 0, 1, 0],
11, [1, 5, 4, 3, 2, 2, 3, 4, 5, 1, 0],
12, [1, 0, 0, 0, 5, 0, 5, 0, 0, 0, 1, 0],
...
Row 5 of A347684 starts 1, 9, 11, 9, 0, 11, 29, 31, 19, 0, 21, 49, 51, 29, 0, 31, ... Subtracting the first block of five terms from the second block of five terms we get [11, 29, 31, 19, 0] - [1, 9, 11, 9, 0] = [10, 20, 20, 10, 0], and after dividing by 10 we get [1, 2, 2, 1, 0], which is row 5 of the present triangle.

N. J. A. Sloane, Table of n, a(n) for n = 1..5050 [Rows 1 through 100, flattened]
N. J. A. Sloane, Rows 1 through 40

Cf. A347681, A347682, A347683, A347684.

myfun1 := proc(A,B) local Ar,Br;
if igcd(A,B) > 1 then return(0); fi;
  Ar:=(A)^(-1) mod B;
   if 2*Ar > B then Ar:=B-Ar; fi;
  Br:=(B)^(-1) mod A;
   if 2*Br > A then Br:=A-Br; fi;
A*Ar+B*Br;
end;
T87:=(n,k) -> (myfun1(n,n+k)-myfun1(n,k))/(2*n);
for n from 1 to 20 do lprint([seq(T87(n,k),k=1..n)]); od:

A347681 Triangle read by rows: T(n,k) (1<=k<=n) = f(prime(n),prime(k)), where f(x,y) = xred_inv(x,y) + yred_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.

Original entry on oeis.org

0, 5, 0, 9, 11, 0, 13, 13, 29, 0, 21, 23, 21, 43, 0, 25, 25, 51, 27, 131, 0, 33, 35, 69, 69, 67, 103, 0, 37, 37, 39, 113, 153, 77, 305, 0, 45, 47, 91, 139, 45, 183, 137, 229, 0, 57, 59, 59, 57, 175, 233, 407, 115, 231, 0, 61, 61, 61, 125, 309, 311, 373, 495, 185, 869, 0, 73, 73, 149, 223, 221, 443, 443, 75, 369, 813, 371, 0

Offset: 1

N. J. A. Sloane, Sep 18 2021

If u, v are positive integers with gcd(u,v) = 1, the "reduced inverse" red_inv(u,v) of u mod v is u^(-1) mod v if u^(-1) mod v <= v/2, otherwise it is v - u^(-1) mod v.
That is, we map u to whichever of +-u has a representative mod v in the range 0 to v/2. Stated another way, red_inv(u,v) is a number r in the range 0 to v/2 such that r*u == +-1 mod v.
For example, red_inv(3,11) = 4, since 3^(-1) mod 11 = 4. But red_inv(2,11) = 5 = 11-6, since red_inv(2,11) = 6.
Arises in the study of A344005.

			Triangle begins:
0,
5, 0,
9, 11, 0,
13, 13, 29, 0,
21, 23, 21, 43, 0,
25, 25, 51, 27, 131, 0,
33, 35, 69, 69, 67, 103, 0,
37, 37, 39, 113, 153, 77, 305, 0,
45, 47, 91, 139, 45, 183, 137, 229, 0,
57, 59, 59, 57, 175, 233, 407, 115, 231, 0,
...

N. J. A. Sloane, Table of n, a(n) for n = 1..5050 [First 100 rows, flattened]

Cf. A344005, A347682, A347683, A347684.

myfun1 := proc(A,B) local Ar,Br;
if igcd(A,B) > 1 then return(0); fi;
  Ar:=(A)^(-1) mod B;
   if 2*Ar > B then Ar:=B-Ar; fi;
  Br:=(B)^(-1) mod A;
   if 2*Br > A then Br:=A-Br; fi;
A*Ar+B*Br;
end;
myfun2:=(i,j)->myfun1(ithprime(i),ithprime(j));
for i from 1 to 20 do lprint([seq(myfun2(i,j),j=1..i)]); od:

A347682 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = f(prime(n),prime(k)), where f(x,y) = xred_inv(x,y) + yred_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.

Original entry on oeis.org

0, 5, 5, 9, 0, 9, 13, 11, 11, 13, 21, 13, 0, 13, 21, 25, 23, 29, 29, 23, 25, 33, 25, 21, 0, 21, 25, 33, 37, 35, 51, 43, 43, 51, 35, 37, 45, 37, 69, 27, 0, 27, 69, 37, 45, 57, 47, 39, 69, 131, 131, 69, 39, 47, 57, 61, 59, 91, 113, 67, 0, 67, 113, 91, 59, 61, 73, 61, 59, 139, 153, 103, 103, 153, 139, 59, 61, 73

Offset: 1

N. J. A. Sloane, Sep 18 2021

If u, v are positive integers with gcd(u,v) = 1, the "reduced inverse" red_inv(u,v) of u mod v is u^(-1) mod v if u^(-1) mod v <= v/2, otherwise it is v - u^(-1) mod v.
That is, we map u to whichever of +-u has a representative mod v in the range 0 to v/2. Stated another way, red_inv(u,v) is a number r in the range 0 to v/2 such that r*u == +-1 mod v.
For example, red_inv(3,11) = 4, since 3^(-1) mod 11 = 4. But red_inv(2,11) = 5 = 11-6, since red_inv(2,11) = 6.
Arises in the study of A344005.

			The array begins:
0, 5, 9, 13, 21, 25, 33, 37, 45, 57, 61, 73,...
5, 0, 11, 13, 23, 25, 35, 37, 47, 59, 61, 73,...
9, 11, 0, 29, 21, 51, 69, 39, 91, 59, 61, 149,...
13, 13, 29, 0, 43, 27, 69, 113, 139, 57, 125, 223,...
21, 23, 21, 43, 0, 131, 67, 153, 45, 175, 309, 221,...
25, 25, 51, 27, 131, 0, 103, 77, 183, 233, 311, 443,...
33, 35, 69, 69, 67, 103, 0, 305, 137, 407, 373, 443,...
37, 37, 39, 113, 153, 77, 305, 0, 229, 115, 495, 75,...
...
The first few antidiagonals are:
[0]
[5, 5]
[9, 0, 9]
[13, 11, 11, 13]
[21, 13, 0, 13, 21]
[25, 23, 29, 29, 23, 25]
[33, 25, 21, 0, 21, 25, 33]
[37, 35, 51, 43, 43, 51, 35, 37]
...

N. J. A. Sloane, Table of n, a(n) for n = 1..5050 [First 100 antidiagonals, flattened]

Cf. A344005, A347681, A347683, A347684.

Rows 1 and 2 are (essentially) A076274 and A208296.

myfun1 := proc(A,B) local Ar,Br;
if igcd(A,B) > 1 then return(0); fi;
  Ar:=(A)^(-1) mod B;
   if 2*Ar > B then Ar:=B-Ar; fi;
  Br:=(B)^(-1) mod A;
   if 2*Br > A then Br:=A-Br; fi;
A*Ar+B*Br;
end;
myfun2:=(i,j)->myfun1(ithprime(i),ithprime(j));
for i from 1 to 30 do lprint([seq(myfun2(i-j+1,j),j=1..i)]); od:

A347683 Triangle read by rows: T(n,k) (1<=k<=n) = f(n,k), where f(x,y) = xred_inv(x,y) + yred_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.

Original entry on oeis.org

0, 1, 0, 1, 5, 0, 1, 0, 7, 0, 1, 9, 11, 9, 0, 1, 0, 0, 0, 11, 0, 1, 13, 13, 15, 29, 13, 0, 1, 0, 17, 0, 31, 0, 15, 0, 1, 17, 0, 17, 19, 0, 55, 17, 0, 1, 0, 19, 0, 0, 0, 41, 0, 19, 0, 1, 21, 23, 23, 21, 23, 43, 65, 89, 21, 0, 1, 0, 0, 0, 49, 0, 71, 0, 0, 0, 23, 0, 1, 25, 25, 25, 51, 25, 27, 79, 53, 79, 131, 25, 0

Offset: 1

N. J. A. Sloane, Sep 18 2021

If u, v are positive integers with gcd(u,v) = 1, the "reduced inverse" red_inv(u,v) of u mod v is u^(-1) mod v if u^(-1) mod v <= v/2, otherwise it is v - u^(-1) mod v.
That is, we map u to whichever of +-u has a representative mod v in the range 0 to v/2. Stated another way, red_inv(u,v) is a number r in the range 0 to v/2 such that r*u == +-1 mod v.
For example, red_inv(3,11) = 4, since 3^(-1) mod 11 = 4. But red_inv(2,11) = 5 = 11-6, since red_inv(2,11) = 6.
Arises in the study of A344005.

			Triangle begins:
0,
1, 0,
1, 5, 0,
1, 0, 7, 0,
1, 9, 11, 9, 0,
1, 0, 0, 0, 11, 0,
1, 13, 13, 15, 29, 13, 0,
1, 0, 17, 0, 31, 0, 15, 0,
1, 17, 0, 17, 19, 0, 55, 17, 0,
1, 0, 19, 0, 0, 0, 41, 0, 19, 0,
...

N. J. A. Sloane, Table of n, a(n) for n = 1..5050 [The first 100 rows, flattened]

Cf. A344005, A347681, A347682, A347684.

myfun1 := proc(A,B) local Ar,Br;
if igcd(A,B) > 1 then return(0); fi;
  Ar:=(A)^(-1) mod B;
   if 2*Ar > B then Ar:=B-Ar; fi;
  Br:=(B)^(-1) mod A;
   if 2*Br > A then Br:=A-Br; fi;
A*Ar+B*Br;
end;
for i from 1 to 20 do lprint([seq(myfun1(i,j),j=1..i)]); od:

cp's OEIS Frontend

A347687 Triangle read by rows: T(n,k) (1<=k<=n) = (r(n+k)-r(k))/(2*n), where {r(i), i>=1} is the n-th row of array A347684.

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A347681 Triangle read by rows: T(n,k) (1<=k<=n) = f(prime(n),prime(k)), where f(x,y) = x*red_inv(x,y) + y*red_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.

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Comments

Examples

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Crossrefs

Programs

Maple

A347682 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = f(prime(n),prime(k)), where f(x,y) = x*red_inv(x,y) + y*red_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A347683 Triangle read by rows: T(n,k) (1<=k<=n) = f(n,k), where f(x,y) = x*red_inv(x,y) + y*red_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A347681 Triangle read by rows: T(n,k) (1<=k<=n) = f(prime(n),prime(k)), where f(x,y) = xred_inv(x,y) + yred_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.

A347682 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = f(prime(n),prime(k)), where f(x,y) = xred_inv(x,y) + yred_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.

A347683 Triangle read by rows: T(n,k) (1<=k<=n) = f(n,k), where f(x,y) = xred_inv(x,y) + yred_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.