A054431 Array read by antidiagonals: T(x, y) tells whether (x, y) are coprime (1) or not (0).

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1

Offset: 1

Author

Antti Karttunen

Keywords

• nonn
• tabl

Comments

Array is read along (x, y) = (1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1), ...

There are nontrivial infinite paths of 1's in this sequence, moving only 1 step down or to the right at each step. Starting at (1,1), move down to (2,1), then (3,1), ..., (13,1). Then move right to (13,2), (13,3), ..., (13,11). From this point, alternate moving down to the next prime row, and right to the next prime column. - Franklin T. Adams-Watters, May 27 2014

Examples

			Rows start:
  1, 1, 1, 1, 1, 1, ...;
  1, 0, 1, 0, 1, 0, ...;
  1, 1, 0, 1, 1, 0, ...;
  1, 0, 1, 0, 1, 0, ...;
  1, 1, 1, 1, 0, 1, ...;
  1, 0, 0, 0, 1, 0, ...;
		

Crossrefs

Equal to A003989 with non-one values replaced with zeros.

Cf. A047999, A054432, A055088, A054521, A215200.

Programs

Maple

reduced_residue_set_0_1_array := n -> one_or_zero(igcd(((n-((trinv(n)*(trinv(n)-1))/2))+1), ((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1) ));
one_or_zero := n -> `if`((1 = n),(1),(0)); # trinv given at A054425
A054431_row := n -> seq(abs(numtheory[jacobi](n-k+1,k)),k=1..n);
for n from 1 to 14 do A054431_row(n) od; # Peter Luschny, Aug 05 2012

Mathematica

t[n_, k_] := Boole[CoprimeQ[n, k]]; Table[t[n-k+1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 21 2012 *)

Sage

def A054431_row(n): return [abs(kronecker_symbol(n-k+1,k)) for k in (1..n)]
for n in (1..14): print(A054431_row(n)) # Peter Luschny, Aug 05 2012

Formula

T(n, k) = T(n, k-n) + T(n-k, k) starting with T(n, k)=0 if n or k are nonpositive and T(1, 1)=1. T(n, k) = A054521(n, k) if n>=k, = A054521(k, n) if n<=k. Antidiagonal sums are phi(n) = A000010(n). - Henry Bottomley, May 14 2002

As a triangular array for n>=1, 1<=k<=n, T(n,k) = |K(n-k+1|k)| where K(i|j) is the Kronecker symbol. - Peter Luschny, Aug 05 2012

Dirichlet g.f.: Sum_{n>=1} Sum_{k>=1} [gcd(n,k)=1]/n^s/k^c = zeta(s)*zeta(c)/zeta(s + c). - Mats Granvik, May 19 2021