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.

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

Author

N. J. A. Sloane, Sep 18 2021

Keywords

• nonn
• tabl

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.

Examples

			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,
...
		

Links

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

Crossrefs

Cf. A344005, A347682, A347683, A347684.

Programs

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;
myfun2:=(i,j)->myfun1(ithprime(i),ithprime(j));
for i from 1 to 20 do lprint([seq(myfun2(i,j),j=1..i)]); od: