A057567 Number of partitions of n where the product of parts divides n.

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

Offset: 1

Author

Leroy Quet, Oct 04 2000

Keywords

• nonn

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

Examples

			From _Gus Wiseman_, Jul 04 2019: (Start)
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)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000070, A000110, A001055 (Mobius transform), A002110, A025487, A057568, A108464, A113309, A131802.

Any prime numbered column of array A108461.

Cf. A028422, A096276, A114324, A318950, A319000, A319005, A326152, A326155.

Programs

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 *)

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}
A001055(n) = fcnt(n, n) \\ This function from Michael B. Porter, Oct 29 2009
A057567(n) = sumdiv(n, d, A001055(d)); \\ After Jovovic's formula. Antti Karttunen, May 25 2017

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))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 19 2017

Formula

a(n) = Sum_{d|n} A001055(d). - Vladeta Jovovic, Nov 19 2000

a(A025487(n)) = A108464(n).

a(p^k) = A000070(k).

a(A002110(n)) = A000110(n+1).

Dirichlet g.f.: zeta(s) * Product_{k>=2} 1/(1 - 1/k^s). - Ilya Gutkovskiy, Nov 03 2020

Extensions

More terms from James Sellers, Oct 09 2000

More terms from Vladeta Jovovic, Nov 19 2000