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

1,2

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 [First 100 rows, flattened]

Example

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,
...

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:

Crossrefs

Cf. A344005, A347682, A347683, A347684.

Sequence in context: A176325 A275792 A010481 * A374285 A291724 A340950

Adjacent sequences: A347678 A347679 A347680 * A347682 A347683 A347684

Keyword

nonn,tabl

Author

N. J. A. Sloane, Sep 18 2021

Status

approved