A140256 Triangle read by columns: Column k is A014963 aerated with groups of (k-1) zeros.

1, 2, 1, 3, 0, 1, 2, 2, 0, 1, 5, 0, 0, 0, 1, 1, 3, 2, 0, 0, 1, 7, 0, 0, 0, 0, 0, 1, 2, 2, 0, 2, 0, 0, 0, 1, 3, 0, 3, 0, 0, 0, 0, 0, 1, 1, 5, 0, 0, 2, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 0, 2, 0, 0, 0, 0, 0, 1, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 7, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Gary W. Adamson and Mats Granvik, May 16 2008, Jun 11 2008

Keywords

• nonn
• tabl

Comments

If the row number n is prime, the row consists of T(n,1)=n followed by n-2 zeros and followed by T(n,n)=1.

Similar to A138618.

Row products of nonzero terms in row n, equals n. - Mats Granvik, May 22 2016

Examples

			First few rows of the triangle are:
   1;
   2, 1;
   3, 0, 1;
   2, 2, 0, 1;
   5, 0, 0, 0, 1;
   1, 3, 2, 0, 0, 1;
   7, 0, 0, 0, 0, 0, 1;
   2, 2, 0, 2, 0, 0, 0, 1;
   3, 0, 3, 0, 0, 0, 0, 0, 1;
   1, 5, 0, 0, 2, 0, 0, 0, 0, 1;
  11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
   1, 1, 2, 3, 0, 2, 0, 0, 0, 0, 0, 1;
  ...
Column 2 = (1, 0, 2, 0, 3, 0, 2, 0, 5, 0, 1, 0, 7, ...).
		

Links

• T. Tao, Simons Lecture I: Structure and randomness in Fourier analysis and number theory.
• Wikipedia, Fundamental theorem of arithmetic.

Crossrefs

Cf. A140255 (row sums), A014963.

Row products without the zero terms produce A000027. [Mats Granvik, Oct 08 2009]

Programs

Excel

=if(row()>=column();if(mod(row();column())=0;lookup(roundup(row()/column();0);A000027;A014963);0);"")

Mathematica

t[n_, k_] /; Divisible[n, k] := Exp[ MangoldtLambda[n/k] ]; t[, ] = 0; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 28 2013 *)
(* recurrence *)
Clear[t, s, n, k, z, nn];z = 1;nn = 14;t[n_, k_] := t[n, k] = If[k == 1, Zeta[s]*(1 - 1/n^(s - 1)) -Sum[t[n, i]/i^(s - 1), {i, 2, n}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; A = Table[Table[Limit[t[n, k], s -> z], {k, 1, n}], {n, 1, nn}]; Flatten[Exp[A]*Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Apr 09 2016, May 22 2016 *)

Formula

T(n,k) = A014963(n/k) = A014963(A126988(n,k)) if k|n, T(n,k)=0 otherwise. 1 <= k <= n.

From Mats Granvik, Apr 10 2016, May 22 2016: (Start)

Limit as s -> 1 of the recurrence: Ts(n, k) = if k = 1 then zeta(s)*(1 - 1/n^(s - 1)) -Sum_{i=2..n} Ts(n, i)/(i)^(s - 1) else if n mod k = 0 then Ts(n/k, 1) else 0 else 0.

For n not equal to k: Limit as s -> 1 of the recurrence: Ts(n, k) = if k = 1 then zeta(s) -Sum_{i=2..n} Ts(n, i)/i^(s - 1) else if n mod k = 0 then Ts(n/k, 1) else 0 else 0.

Limit as s -> 1 of the recurrence: Ts(n, k) = if k = 1 then log(n) -Sum_{i=2..n} Ts(n, i)/i^(s - 1) else if n mod k = 0 then Ts(n/k, 1) else 0 else 0. (End)

[The above sentences need a lot of work! Parentheses might help. - N. J. A. Sloane, Mar 14 2017]