A175992 Triangle T(n,k) read by rows. If n=k then 0, else if k divides n then 1 else 0.

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

Offset

1

Comments

Obtained by replacing the elements on the diagonal of A051731 by zero.

Apart from the initial T(1,1), the same as A176890.

The inverse of A051731 can be calculated with matrix powers of this matrix T(n,k) as follows: T(n,k)^0-T(n,k)^1+T(n,k)^2-T(n,k)^3+T(n,k)^4 ... Compare this to the binomial series for (1+x)^-1=1-x+x^2-x^3+x^4...

Links

Table of n, a(n) for n=1..190.

Formula

T(n,n) = 0. T(n,k) = 1 if k|n and k<n. T(n,k) = 0 if k does not divide n and k<n.

Lim_{s -> Infinity} T(n,k) where Ts(n,k) is the table defined by the recurrence:

Ts(n,k)=if k=1 then zeta[s]*(1-1/n^(s-1))-Sum_{i=1..n-1} Ts(n,k+i)/(i+1)^(s-1) else if k divides n then Ts(n/k,1) else Ts(n,k)=0 else Ts(n,k)=0. - Mats Granvik, Apr 09 2016

Example

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

Mathematica

Flatten[Table[Table[If[Mod[n, k] == 0, If[n == k, 0, 1], 0], {k, 1, n}], {n, 1, 19}]] (* Mats Granvik, May 25 2013 *)
Flatten[Table[Which[n==k, 0, Divisible[n, k], 1, True, 0], {n, 20}, {k, n}]] (* Harvey P. Dale, Oct 03 2014 *)
(* recurrence *)
Clear[t, s, n, k, z, nn]; nn = 19; t[n_, k_] := t[n, k] = If[k == 1, Zeta[s]*(1 - 1/n^(s - 1)) -Sum[t[n, k + i]/(i + 1)^(s - 1), {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; Flatten[Table[Table[Limit[t[n, k], s -> Infinity], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Apr 09 2016 *)

Crossrefs

Cf. A051731, A176890, A008683.

Sequence in context: A296657 A082848 A078588 * A173922 A213676 A309752

Adjacent sequences: A175989 A175990 A175991 * A175993 A175994 A175995

Keyword

nonn,tabl,easy

Author

Mats Granvik, Dec 06 2010

Status

approved