This site is supported by donations to The OEIS Foundation.

 Annual Appeal: Please make a donation to keep the OEIS running. In 2018 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A140256 Triangle read by columns: Column k is A014963 aerated with groups of (k-1) zeros. 7
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 LINKS Wikipedia, Fundamental theorem of arithmetic. 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] EXAMPLE 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, ...). 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 *) PROG (Excel) =if(row()>=column(); if(mod(row(); column())=0; lookup(roundup(row()/column(); 0); A000027; A014963); 0); "") CROSSREFS Cf. A140255 (row sums), A014963. Row products without the zero terms produce A000027. [Mats Granvik, Oct 08 2009] Sequence in context: A182740 A228786 A140699 * A126206 A119709 A253556 Adjacent sequences:  A140253 A140254 A140255 * A140257 A140258 A140259 KEYWORD nonn,tabl AUTHOR Gary W. Adamson and Mats Granvik, May 16 2008, Jun 11 2008 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 11 10:48 EST 2018. Contains 318049 sequences. (Running on oeis4.)