A054431 - OEIS

%I #36 May 23 2021 10:28:55

%S 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,

%T 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,

%U 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

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

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

%C 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

%F 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

%F 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

%F 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

%e Rows start:

%e   1, 1, 1, 1, 1, 1, ...;

%e   1, 0, 1, 0, 1, 0, ...;

%e   1, 1, 0, 1, 1, 0, ...;

%e   1, 0, 1, 0, 1, 0, ...;

%e   1, 1, 1, 1, 0, 1, ...;

%e   1, 0, 0, 0, 1, 0, ...;

%p 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) ));

%p one_or_zero := n -> `if`((1 = n),(1),(0)); # trinv given at A054425

%p A054431_row := n -> seq(abs(numtheory[jacobi](n-k+1,k)),k=1..n);

%p for n from 1 to 14 do A054431_row(n) od; # _Peter Luschny_, Aug 05 2012

%t 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 *)

%o (Sage)

%o def A054431_row(n): return [abs(kronecker_symbol(n-k+1,k)) for k in (1..n)]

%o for n in (1..14): print(A054431_row(n)) # _Peter Luschny_, Aug 05 2012

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

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

%K nonn,tabl

%O 1,1

%A _Antti Karttunen_