|
|
A057567
|
|
Number of partitions of n where the product of parts divides n.
|
|
37
|
|
|
1, 2, 2, 4, 2, 5, 2, 7, 4, 5, 2, 11, 2, 5, 5, 12, 2, 11, 2, 11, 5, 5, 2, 21, 4, 5, 7, 11, 2, 15, 2, 19, 5, 5, 5, 26, 2, 5, 5, 21, 2, 15, 2, 11, 11, 5, 2, 38, 4, 11, 5, 11, 2, 21, 5, 21, 5, 5, 2, 36, 2, 5, 11, 30, 5, 15, 2, 11, 5, 15, 2, 52, 2, 5, 11, 11, 5, 15, 2, 38, 12, 5, 2, 36, 5, 5, 5, 21
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1). - Christian G. Bower, Jun 03 2005
|
|
LINKS
|
|
|
FORMULA
|
Dirichlet g.f.: zeta(s) * Product_{k>=2} 1/(1 - 1/k^s). - Ilya Gutkovskiy, Nov 03 2020
|
|
EXAMPLE
|
The a(1) = 1 through a(9) = 5 partitions are the following. The Heinz numbers of these partitions are given by A326155.
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (11111) (321) (1111111) (4211)
(211) (3111) (22211)
(1111) (21111) (41111)
(111111) (221111)
(2111111)
(11111111)
(End)
|
|
MATHEMATICA
|
Table[Function[m, Count[Map[Times @@ # &, IntegerPartitions[m]], P_ /; Divisible[m, P]] - Boole[n == 1]]@ Apply[Times, #] &@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]], {n, 88}] (* Michael De Vlieger, Aug 16 2017 *)
|
|
PROG
|
(PARI)
fcnt(n, m) = {local(s); s=0; if(n == 1, s=1, fordiv(n, d, if(d > 1 & d <= m, s=s+fcnt(n/d, d)))); s}
(Python)
from sympy import divisors, isprime
def T(n, m):
if isprime(n): return 1 if n <= m else 0
A = (d for d in divisors(n) if 1 < d < n and d <= m)
s = sum(T(n // d, d) for d in A)
return s + 1 if n <= m else s
def a001055(n): return T(n, n)
def a(n): return sum(a001055(d) for d in divisors(n))
|
|
CROSSREFS
|
Any prime numbered column of array A108461.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|