cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A181844 Sum over all partitions of n of the LCM of the parts.

Original entry on oeis.org

1, 1, 3, 6, 12, 23, 38, 73, 118, 198, 318, 530, 819, 1298, 1974, 2975, 4516, 6698, 9980, 14550, 21186, 30304, 43503, 62030, 87908, 123292, 172543, 239720, 331688, 458198, 629376, 860332, 1168172, 1583176, 2138438, 2876283, 3859770, 5159886, 6863702, 9112356
Offset: 0

Views

Author

Peter Luschny, Dec 07 2010

Keywords

Comments

Old name was: Row sums of A181842.

Links

Crossrefs

Cf. A078392 (the same for GCD), A181843, A181842, A256067, A256553, A256554, A306956.

Programs

  • Maple
    with(combstruct):
a181844 := proc(n) local k,L,l,R,part;
R := NULL; L := 0;
for k from 1 to n do
   part := iterstructs(Partition(n),size=k):
   while not finished(part) do
      l := nextstruct(part);
      L := L + ilcm(op(l));
   od;
od;
L end:
# second Maple program:
b:= proc(n, i, r) option remember; `if`(n=0, r, `if`(i<1, 0,
       b(n, i-1, r)+b(n-i, min(i, n-i), ilcm(i, r))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..42);  # Alois P. Heinz, Mar 18 2019
  • Mathematica
    t[n_, k_] := LCM @@@ IntegerPartitions[n, {n - k + 1}] // Total; a[n_] := Sum[t[n, k], {k, 1, n}]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Jul 26 2013 *)

Formula

a(n) = Sum_{k>=0} k * A256067(n,k) = Sum_{k>=0} A256553(n,k)*A256554(n,k). - Alois P. Heinz, Apr 02 2015

Extensions

a(0)=1 prepended by Alois P. Heinz, Mar 29 2015
New name from Alois P. Heinz, Mar 18 2019

A181842 Triangle read by rows: T(n,k) = Sum_{c in partition(n,n-k+1)} lcm(c).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 5, 10, 5, 1, 2, 5, 12, 12, 6, 1, 2, 5, 12, 18, 28, 7, 1, 2, 5, 12, 20, 38, 32, 8, 1, 2, 5, 12, 20, 44, 57, 48, 9, 1, 2, 5, 12, 20, 46, 67, 100, 55, 10
Offset: 1

Views

Author

Peter Luschny, Dec 07 2010

Keywords

Comments

In A181842 through A181854 the following terminology is used.
Let n, k be positive integers.
* Partition: A (n,k)-partition is the set of all k-sets of
positive integers whose elements sum to n.
- The cardinality of a (n,k)-partition: A008284(n,k).
- Maple: (n,k) -> combstruct[count](Partition(n),size=k).
- The (6,2)-partition is {{1,5},{2,4},{3,3}}.
* Composition: A (n,k)-composition is the set of all k-tuples of positive integers whose elements sum to n.
- The cardinality of a (n,k)-composition: A007318(n-1,k-1).
- Maple: (n,k) -> combstruct[count](Composition(n),size=k).
- The (6,2)-composition is {<5,1>,<4,2>,<3,3>,<2,4>,<1,5>}.
* Combination: A (n,k)-combination is the set of all k-subsets
of {1,2,..,n}.
- The cardinality of a (n,k)-combination: A007318(n,k).
- Maple: (n,k) -> combstruct[count](Combination(n),size=k).
- The (4,2)-combination is {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}.

Examples

			[1]   1
[2]   1   2
[3]   1   2   3
[4]   1   2   5   4
[5]   1   2   5   10   5
[6]   1   2   5   12   12   6
[7]   1   2   5   12   18   28   7

Crossrefs

Cf. A181843, A181844.

Programs

  • Maple
    with(combstruct):
a181842_row := proc(n) local k,L,l,R,part;
R := NULL;
for k from 1 to n do
   L := 0;
   part := iterstructs(Partition(n),size=n-k+1):
   while not finished(part) do
      l := nextstruct(part);
      L := L + ilcm(op(l));
   od;
   R := R,L;
od;
R end:
  • Mathematica
    t[n_, k_] := LCM @@@ IntegerPartitions[n, {n - k + 1}] // Total; Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 26 2013 *)

A181845 Triangle read by rows: T(n,k) = max_{c in P(n,n-k+1)} lcm(c) where P(n,m) = A008284(n,m) is the number of partitions of n into m parts.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 6, 5, 1, 2, 3, 6, 5, 6, 1, 2, 3, 6, 6, 12, 7, 1, 2, 3, 6, 6, 12, 15, 8, 1, 2, 3, 6, 6, 12, 15, 20, 9, 1, 2, 3, 6, 6, 12, 15, 30, 21, 10, 1, 2, 3, 6, 6, 12, 15, 30, 21, 30, 11, 1, 2, 3, 6, 6, 12, 15, 30, 30, 60, 35, 12
Offset: 1

Views

Author

Peter Luschny, Dec 07 2010

Keywords

Comments

See A181842 for the definition of 'partition'. T(n,k) is also the triangle read by rows: T(n,k) = max_{c in C(n,n-k+1)} lcm(c) where C(n,m) is the set of all m-tuples of positive integers whose elements sum to n where the C(n,k) = A007318(n-1,k-1) are called compositions of n of size k.

Examples

			[1]   1
[2]   1   2
[3]   1   2   3
[4]   1   2   3   4
[5]   1   2   3   6   5
[6]   1   2   3   6   5   6
[7]   1   2   3   6   6   12   7
[8]   1   2   3   6   6   12   15   8
[9]   1   2   3   6   6   12   15   20   9

Links

Crossrefs

Cf. A181842, A181843, A181844.

Programs

  • Maple
    with(combstruct):
a181845_row := proc(n) local k,L,l,R,part;
R := NULL;
for k from 1 to n do
   L := 0;
   part := iterstructs(Partition(n),size=n-k+1):
# alternatively (but slower)
# part := iterstructs(Composition(n), size=n-k+1):
   while not finished(part) do
      l := nextstruct(part);
      L := max(L,ilcm(op(l)));
   od;
   R := R,L;
od;
R end:
  • PARI
    Row(n)={my(v=vector(n)); forpart(p=n, my(i=#p); v[i]=max(v[i], lcm(Vec(p)))); Vecrev(v)}
{ for(n=1, 10, print(Row(n))) } \\ Andrew Howroyd, Apr 20 2021

Extensions

Terms a(56) and beyond from Andrew Howroyd, Apr 20 2021
Showing 1-3 of 3 results.

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