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
Examples
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
Links
- Mats Granvik, Oct 13 2007, Table of n, a(n) for n = 1..406
Programs
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Excel
=if(and(row()>=column();row()
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Excel
=if(and(n>=k; n < A014963*A100994); A100994; 1) - Mats Granvik, Jan 21 2008
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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
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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 *)
Formula
T(n,k) = if nA120112(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
Extensions
Indices added to formulas by R. J. Mathar, Nov 23 2010
Comments