A133232 - OEIS

%I #18 Mar 01 2021 17:54:12

%S 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,

%T 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,

%U 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

%N 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).

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

%H Mats Granvik, Oct 13 2007, <a href="/A133232/b133232.txt">Table of n, a(n) for n = 1..406</a>

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

%F T(n,k) = if n < A014963(k)*A100994(k) then A100994(k), else 1 (1<=k<=n). - _Mats Granvik_, Jan 21 2008

%e 2 occurs 2*1 = 2 times in column 2.

%e 3 occurs 3*2 = 6 times in column 3.

%e 4 occurs 4*1 = 4 times in column 4.

%e 5 occurs 5*4 = 20 times in column 5.

%e k occurs A133936(k) times in column k. The first rows of the triangle and the least common multiple of the rows are:

%e lcm{1} = 1

%e lcm{1, 2} = 2

%e lcm{1, 2, 3} = 6

%e lcm{1, 1, 3, 4} = 12

%e lcm{1, 1, 3, 4, 5} = 60

%e lcm{1, 1, 3, 4, 5, 1} = 60

%e lcm{1, 1, 3, 4, 5, 1, 7} = 420

%e lcm{1, 1, 3, 1, 5, 1, 7, 8} = 840

%e lcm{1, 1, 1, 1, 5, 1, 7, 8, 9} = 2520

%p A120112 := proc(n) 1-ilcm(seq(i,i=1..n+1))/ilcm(seq(i,i=1..n)) ; end proc:

%p A133232 := proc(n) if n < k*(1+abs(A120112(k-1))) then k else 1; end if; end proc:

%p seq(seq(A133232(n,k),k=1..n),n=1..15) ; # _R. J. Mathar_, Nov 23 2010

%t b[n_] := b[n] = If[n == 0, 1, LCM @@ Range[n]];

%t c[n_] := 1 - b[n+1]/b[n];

%t T[n_, k_] := If[n < k*(1+Abs[c[k-1]]), k, 1];

%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 01 2021 *)

%o (Excel) =if(and(row()>=column();row()<column()+column()*abs(A120112));column();1)

%o (Excel) =if(and(n>=k; n < A014963*A100994); A100994; 1) - _Mats Granvik_, Jan 21 2008

%Y Cf. A003418, A120112, A014963, A134579.

%K nonn,tabl

%O 1,3

%A _Mats Granvik_, Oct 13 2007

%E Indices added to formulas by _R. J. Mathar_, Nov 23 2010