cp's OEIS Frontend

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

%I A054431 #36 May 23 2021 10:28:55
%S A054431 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 A054431 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 A054431 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 A054431 Array read by antidiagonals: T(x, y) tells whether (x, y) are coprime (1) or not (0).
%C A054431 Array is read along (x, y) = (1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1), ...
%C A054431 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 A054431 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 A054431 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 A054431 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 A054431 Rows start:
%e A054431   1, 1, 1, 1, 1, 1, ...;
%e A054431   1, 0, 1, 0, 1, 0, ...;
%e A054431   1, 1, 0, 1, 1, 0, ...;
%e A054431   1, 0, 1, 0, 1, 0, ...;
%e A054431   1, 1, 1, 1, 0, 1, ...;
%e A054431   1, 0, 0, 0, 1, 0, ...;
%p A054431 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 A054431 one_or_zero := n -> `if`((1 = n),(1),(0)); # trinv given at A054425
%p A054431 A054431_row := n -> seq(abs(numtheory[jacobi](n-k+1,k)),k=1..n);
%p A054431 for n from 1 to 14 do A054431_row(n) od; # _Peter Luschny_, Aug 05 2012
%t A054431 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 A054431 (Sage)
%o A054431 def A054431_row(n): return [abs(kronecker_symbol(n-k+1,k)) for k in (1..n)]
%o A054431 for n in (1..14): print(A054431_row(n)) # _Peter Luschny_, Aug 05 2012
%Y A054431 Equal to A003989 with non-one values replaced with zeros.
%Y A054431 Cf. A047999, A054432, A055088, A054521, A215200.
%K A054431 nonn,tabl
%O A054431 1,1
%A A054431 _Antti Karttunen_