A094308 -id:A094308 - cp's OEIS frontend

A181853 Triangle read by rows: T(n,k) = Sum_{c in C(n,k)} lcm(c) where C(n,k) is the set of all k-subsets of {1,2,...,n}.

1, 1, 1, 1, 3, 2, 1, 6, 11, 6, 1, 10, 31, 34, 12, 1, 15, 81, 189, 182, 60, 1, 21, 141, 393, 494, 282, 60, 1, 28, 288, 1380, 3245, 3740, 2034, 420, 1, 36, 456, 2716, 8293, 13268, 11338, 4908, 840, 1, 45, 726, 5578, 22207, 47351, 57598, 40602, 15564, 2520

Offset: 0

Peter Luschny, Dec 06 2010

The C(n,k) are also called combinations of n with size k (see A181842).

Main diagonal gives: A003418. Lower diagonal gives: A094308. Column k=1 gives: A000217. - Alois P. Heinz, Jul 29 2013

			[0]   1
[1]   1    1
[2]   1    3     2
[3]   1    6    11     6
[4]   1   10    31    34    12
[5]   1   15    81   189   182    60
[6]   1   21   141   393   494   282   60

Alois P. Heinz, Rows n = 0..46, flattened

Row sums give A226037.

Cf. A065567, A096179, A181854.

with(combstruct):
a181853_row := proc(n) local k,L,l,R,comb;
R := NULL;
for k from 0 to n do
   L := 0;
   comb := iterstructs(Combination(n),size=k):
   while not finished(comb) do
      l := nextstruct(comb);
      L := L + ilcm(op(l));
   od;
   R := R,L;
od;
R end:
# second Maple program:
b:= proc(n, k) option remember; `if`(k=0, [1],
     [`if`(k add(c, c=b(n, k)):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jul 29 2013
# third Maple program:
b:= proc(n, m) option remember; expand(`if`(n=0, m,
      b(n-1, ilcm(m, n))*x+b(n-1, m)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
seq(T(n), n=0..10);  # Alois P. Heinz, Sep 05 2023

t[, 0] = 1; t[n, k_] := Sum[LCM @@ c, {c, Subsets[Range[n], {k}]}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

# (After Alois P. Heinz)
@CachedFunction
def b(n, k):
    if k == 0: return [1]
    w = b(n-1, k) if kPeter Luschny, Jul 29 2013

A094307 The k-th term of the n-th row of the following triangle is the least common multiple of all numbers from 1 to n except k. Sequence contains the triangle by rows.

Original entry on oeis.org

1, 2, 1, 6, 3, 2, 12, 12, 4, 6, 60, 60, 20, 30, 12, 60, 60, 60, 30, 12, 60, 420, 420, 420, 210, 84, 420, 60, 840, 840, 840, 840, 168, 840, 120, 420, 2520, 2520, 2520, 2520, 504, 2520, 360, 1260, 840, 2520, 2520, 2520, 2520, 2520, 2520, 360, 1260, 840, 2520, 27720

Offset: 1

Amarnath Murthy, Apr 29 2004

The leading diagonal and the first column are given by A003418 with a suitable offset 0 or 1.

			Triangle begins:
   1,
   2,  1,
   6,  3,  2,
  12, 12,  4,  6,
  60, 60, 20, 30, 12,
  60, 60, 60, 30, 12, 60;

Cf. A094308.

A094307 := proc(n,k) local a,i ; if n = 1 then RETURN(1) ; elif k > 1 and k < n then a := [seq(i,i=1..k-1),seq(i,i=k+1..n)] ; elif k = n then a := [seq(i,i=1..k-1)] ; else a := [seq(i,i=2..n)] ; fi ; ilcm(op(a)) ; end: for n from 1 to 15 do for k from 1 to n do printf("%d, ",A094307(n,k)) ; od ; od ; # R. J. Mathar, Apr 30 2007

T[n_, k_] := LCM @@ Which[n == 1, {1}, 1 < k < n, Join[Range[k-1], Range[k+1, n]], k == n, Range[k-1], True, Range[2, n]];
Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 20 2020 *)

T(n,k) = lcm(setminus(vector(n, i, i), Set(k)));
tabl(nn) = {for (n=1, nn, for (k=1, n, print1(T(n, k), ", ");); print(););} \\ Michel Marcus, Jun 15 2019

T(n,k) = A003418(n)/p if k = p^m for some m and n < 2*p^m and A003418(n) otherwise. - Charlie Neder, Jun 13 2019

More terms from R. J. Mathar, Apr 30 2007

cp's OEIS Frontend

A181853 Triangle read by rows: T(n,k) = Sum_{c in C(n,k)} lcm(c) where C(n,k) is the set of all k-subsets of {1,2,...,n}.

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