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.

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

1,5

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.

Links

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

Example

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,
...

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

Crossrefs

Cf. A344005, A347681, A347682, A347684.

Sequence in context: A071086 A339208 A198105 * A339209 A277529 A354133

Adjacent sequences: A347680 A347681 A347682 * A347684 A347685 A347686

Keyword

nonn,tabl

Author

N. J. A. Sloane, Sep 18 2021

Status

approved