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.

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

1,8

Comments

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).

Links

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

Example

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,
...

Maple

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:

Crossrefs

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

Row 3 is A147666.

Sequence in context: A279932 A022697 A286838 * A172457 A028277 A046582

Adjacent sequences: A347681 A347682 A347683 * A347685 A347686 A347687

Keyword

nonn,tabl

Author

N. J. A. Sloane, Sep 18 2021

Status

approved