A347682 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = 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, 5, 9, 0, 9, 13, 11, 11, 13, 21, 13, 0, 13, 21, 25, 23, 29, 29, 23, 25, 33, 25, 21, 0, 21, 25, 33, 37, 35, 51, 43, 43, 51, 35, 37, 45, 37, 69, 27, 0, 27, 69, 37, 45, 57, 47, 39, 69, 131, 131, 69, 39, 47, 57, 61, 59, 91, 113, 67, 0, 67, 113, 91, 59, 61, 73, 61, 59, 139, 153, 103, 103, 153, 139, 59, 61, 73

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 antidiagonals, flattened]

Example

The array begins:
0, 5, 9, 13, 21, 25, 33, 37, 45, 57, 61, 73,...
5, 0, 11, 13, 23, 25, 35, 37, 47, 59, 61, 73,...
9, 11, 0, 29, 21, 51, 69, 39, 91, 59, 61, 149,...
13, 13, 29, 0, 43, 27, 69, 113, 139, 57, 125, 223,...
21, 23, 21, 43, 0, 131, 67, 153, 45, 175, 309, 221,...
25, 25, 51, 27, 131, 0, 103, 77, 183, 233, 311, 443,...
33, 35, 69, 69, 67, 103, 0, 305, 137, 407, 373, 443,...
37, 37, 39, 113, 153, 77, 305, 0, 229, 115, 495, 75,...
...
The first few antidiagonals are:
[0]
[5, 5]
[9, 0, 9]
[13, 11, 11, 13]
[21, 13, 0, 13, 21]
[25, 23, 29, 29, 23, 25]
[33, 25, 21, 0, 21, 25, 33]
[37, 35, 51, 43, 43, 51, 35, 37]
...

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 30 do lprint([seq(myfun2(i-j+1, j), j=1..i)]); od:

Crossrefs

Cf. A344005, A347681, A347683, A347684.

Rows 1 and 2 are (essentially) A076274 and A208296.

Sequence in context: A081287 A303715 A204188 * A334383 A019843 A335319

Adjacent sequences: A347679 A347680 A347681 * A347683 A347684 A347685

Keyword

nonn,tabl

Author

N. J. A. Sloane, Sep 18 2021

Status

approved