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%I #41 Mar 14 2017 00:25:13
%S 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,
%T 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,
%U 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
%N Triangle read by columns: Column k is A014963 aerated with groups of (k-1) zeros.
%C 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.
%C Similar to A138618.
%C Row products of nonzero terms in row n, equals n. - _Mats Granvik_, May 22 2016
%H T. Tao, <a href="http://terrytao.wordpress.com/2007/04/05/simons-lecture-i-structure-and-randomness-in-fourier-analysis-and-number-theory/">Simons Lecture I: Structure and randomness in Fourier analysis and number theory</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic">Fundamental theorem of arithmetic</a>.
%F T(n,k) = A014963(n/k) = A014963(A126988(n,k)) if k|n, T(n,k)=0 otherwise. 1 <= k <= n.
%F From _Mats Granvik_, Apr 10 2016, May 22 2016: (Start)
%F 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.
%F 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.
%F 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)
%F [The above sentences need a lot of work! Parentheses might help. - _N. J. A. Sloane_, Mar 14 2017]
%e First few rows of the triangle are:
%e 1;
%e 2, 1;
%e 3, 0, 1;
%e 2, 2, 0, 1;
%e 5, 0, 0, 0, 1;
%e 1, 3, 2, 0, 0, 1;
%e 7, 0, 0, 0, 0, 0, 1;
%e 2, 2, 0, 2, 0, 0, 0, 1;
%e 3, 0, 3, 0, 0, 0, 0, 0, 1;
%e 1, 5, 0, 0, 2, 0, 0, 0, 0, 1;
%e 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 1, 1, 2, 3, 0, 2, 0, 0, 0, 0, 0, 1;
%e ...
%e Column 2 = (1, 0, 2, 0, 3, 0, 2, 0, 5, 0, 1, 0, 7, ...).
%t 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 *)
%t (* recurrence *)
%t 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 *)
%o (Excel) =if(row()>=column();if(mod(row();column())=0;lookup(roundup(row()/column();0);A000027;A014963);0);"")
%Y Cf. A140255 (row sums), A014963.
%Y Row products without the zero terms produce A000027. [_Mats Granvik_, Oct 08 2009]
%K nonn,tabl
%O 1,2
%A _Gary W. Adamson_ and _Mats Granvik_, May 16 2008, Jun 11 2008
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