A133232 Triangle T(n,k) read by rows with a minimum number of prime powers A100994 for which the least common multiple of T(n,1),..,T(n,n) is A003418(n).

1, 1, 2, 1, 2, 3, 1, 1, 3, 4, 1, 1, 3, 4, 5, 1, 1, 3, 4, 5, 1, 1, 1, 3, 4, 5, 1, 7, 1, 1, 3, 1, 5, 1, 7, 8, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 13, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1

Offset

1,3

Comments

Checked up to 28th row. The rest of the ones in the table are there for the least common multiple to calculate correctly.

Links

Mats Granvik, Oct 13 2007, Table of n, a(n) for n = 1..406

Formula

T(n,k) = if n<k+k*|A120112(k-1)| then k, else 1 (1<=k<=n).

T(n,k) = if n < A014963(k)*A100994(k) then A100994(k), else 1 (1<=k<=n). - Mats Granvik, Jan 21 2008

Example

2 occurs 2*1 = 2 times in column 2.
3 occurs 3*2 = 6 times in column 3.
4 occurs 4*1 = 4 times in column 4.
5 occurs 5*4 = 20 times in column 5.
k occurs A133936(k) times in column k. The first rows of the triangle and the least common multiple of the rows are:
lcm{1} = 1
lcm{1, 2} = 2
lcm{1, 2, 3} = 6
lcm{1, 1, 3, 4} = 12
lcm{1, 1, 3, 4, 5} = 60
lcm{1, 1, 3, 4, 5, 1} = 60
lcm{1, 1, 3, 4, 5, 1, 7} = 420
lcm{1, 1, 3, 1, 5, 1, 7, 8} = 840
lcm{1, 1, 1, 1, 5, 1, 7, 8, 9} = 2520

Maple

A120112 := proc(n) 1-ilcm(seq(i, i=1..n+1))/ilcm(seq(i, i=1..n)) ; end proc:
A133232 := proc(n) if n < k*(1+abs(A120112(k-1))) then k else 1; end if; end proc:
seq(seq(A133232(n, k), k=1..n), n=1..15) ; # R. J. Mathar, Nov 23 2010

Mathematica

b[n_] := b[n] = If[n == 0, 1, LCM @@ Range[n]];
c[n_] := 1 - b[n+1]/b[n];
T[n_, k_] := If[n < k*(1+Abs[c[k-1]]), k, 1];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 01 2021 *)

Prog

(Excel) =if(and(row()>=column(); row()<column()+column()*abs(A120112)); column(); 1)
(Excel) =if(and(n>=k; n < A014963*A100994); A100994; 1) - Mats Granvik, Jan 21 2008

Crossrefs

Cf. A003418, A120112, A014963, A134579.

Sequence in context: A308967 A241844 A233864 * A137152 A328902 A159335

Adjacent sequences: A133229 A133230 A133231 * A133233 A133234 A133235

Keyword

nonn,tabl

Author

Mats Granvik, Oct 13 2007

Extensions

Indices added to formulas by R. J. Mathar, Nov 23 2010

Status

approved