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A355747
Number of multisets that can be obtained by choosing a divisor of each positive integer from 1 to n.
9
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
0,3
FORMULA
a(n) = A355733(A070826(n)).
a(p) = 2*a(p-1) for p prime. - Michael S. Branicky, Aug 03 2022
EXAMPLE
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}
MATHEMATICA
Table[Length[Union[Sort/@Tuples[Divisors/@Range[n]]]], {n, 0, 10}]
PROG
(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
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.
Sequence in context: A104434 A104433 A104432 * A349430 A129211 A175204
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 20 2022
EXTENSIONS
a(15)-a(21) from Michael S. Branicky, Aug 03 2022
a(22)-a(23) from Michael S. Branicky, Aug 08 2022
STATUS
approved