A347687 -id:A347687 - cp's OEIS frontend

A347684 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = 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.

0, 1, 1, 1, 0, 1, 1, 5, 5, 1, 1, 0, 0, 0, 1, 1, 9, 7, 7, 9, 1, 1, 0, 11, 0, 11, 0, 1, 1, 13, 0, 9, 9, 0, 13, 1, 1, 0, 13, 0, 0, 0, 13, 0, 1, 1, 17, 17, 15, 11, 11, 15, 17, 17, 1, 1, 0, 0, 0, 29, 0, 29, 0, 0, 0, 1, 1, 21, 19, 17, 31, 13, 13, 31, 17, 19, 21, 1, 1, 0, 23, 0, 19, 0, 0, 0, 19, 0, 23, 0, 1

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.
Conjecture: The entries of this array can be expressed in terms of A347683 and A347687 as follows. Write k = j*n+i, where j >= 0 and 1 <= i <= n. Then T(n,k) = A347683(n,i) + 2*n*j*A347687(n,i).  For example, suppose n=3, k=11, so k = 3*3+2, with j=3, i=2. Then A347683(3,2) + 2*3*3*A347687(3,2) = 5 + 2*3*3*1 = 23, which is indeed T(3,11).

			The array begins:
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
1, 0, 5, 0, 9, 0, 13, 0, 17, 0, 21, 0, 25, 0, 29, 0,...
1, 5, 0, 7, 11, 0, 13, 17, 0, 19, 23, 0, 25, 29, 0, 31,...
1, 0, 7, 0, 9, 0, 15, 0, 17, 0, 23, 0, 25, 0, 31, 0,...
1, 9, 11, 9, 0, 11, 29, 31, 19, 0, 21, 49, 51, 29, 0, 31,...
1, 0, 0, 0, 11, 0, 13, 0, 0, 0, 23, 0, 25, 0, 0, 0,...
1, 13, 13, 15, 29, 13, 0, 15, 55, 41, 43, 71, 27, 0, 29, 97,...
1, 0, 17, 0, 31, 0, 15, 0, 17, 0, 65, 0, 79, 0, 31, 0,...
1, 17, 0, 17, 19, 0, 55, 17, 0, 19, 89, 0, 53, 55, 0, 127,...
...
The first few antidiagonals are:
0,
1, 1,
1, 0, 1,
1, 5, 5, 1,
1, 0, 0, 0, 1,
1, 9, 7, 7, 9, 1,
1, 0, 11, 0, 11, 0, 1,
1, 13, 0, 9, 9, 0, 13, 1,
1, 0, 13, 0, 0, 0, 13, 0, 1,
1, 17, 17, 15, 11, 11, 15, 17, 17, 1,
...

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

Cf. A344005, A347681, A347682, A347683, A347687.

Row 3 is A147666.

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 14 do lprint([seq(myfun1(i-j+1,j),j=1..i)]); od:

cp's OEIS Frontend

A347684 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

cp's OEIS Frontend

A347684 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = 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

A347684 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = 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.