A355747 Number of multisets that can be obtained by choosing a divisor of each positive integer from 1 to n.

1, 1, 2, 4, 10, 20, 58, 116, 320, 772, 2170, 4340, 14112, 28224, 78120, 212004, 612232, 1224464, 3873760, 7747520, 24224608, 64595088, 175452168, 350904336

Offset: 0

Author

Gus Wiseman, Jul 20 2022

Keywords

• nonn
• more

Examples

			The a(0) = 1 through a(4) = 10 multisets:
  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}
           {1,2}  {1,1,2}  {1,1,1,2}
                  {1,1,3}  {1,1,1,3}
                  {1,2,3}  {1,1,1,4}
                           {1,1,2,2}
                           {1,1,2,3}
                           {1,1,2,4}
                           {1,1,3,4}
                           {1,2,2,3}
                           {1,2,3,4}
		

Crossrefs

The sum of the same integers is A000096.

The product of the same integers is A000142, Heinz number A070826.

Counting sequences instead of multisets gives A066843.

The integers themselves are the rows of A131818 (shifted).

For prime indices we have A355733, only prime factors A355744.

For prime factors instead of divisors we have A355746, factors A355537.

A000005 counts divisors.

A000040 lists the prime numbers.

A001221 counts distinct prime factors, with sum A001414.

A001222 counts prime factors with multiplicity.

Cf. A000720, A002110, A006218, A076610, A327486, A355538, A355731, A355737, A355741, A355745.

Programs

Mathematica

Table[Length[Union[Sort/@Tuples[Divisors/@Range[n]]]],{n,0,10}]

Python

from sympy import divisors
from itertools import count, islice
def agen():
    s = {tuple()}
    for n in count(1):
        yield len(s)
        s = set(tuple(sorted(t+(d,))) for t in s for d in divisors(n))
print(list(islice(agen(), 16))) # Michael S. Branicky, Aug 03 2022

Formula

a(n) = A355733(A070826(n)).

a(p) = 2*a(p-1) for p prime. - Michael S. Branicky, Aug 03 2022

Extensions

a(15)-a(21) from Michael S. Branicky, Aug 03 2022

a(22)-a(23) from Michael S. Branicky, Aug 08 2022