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
a(n) = Sum_{d|n} A001055(d). - Vladeta Jovovic, Nov 19 2000
a(p^k) = A000070(k).
Dirichlet g.f.: zeta(s) * Product_{k>=2} 1/(1 - 1/k^s). - Ilya Gutkovskiy, Nov 03 2020
EXAMPLE
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)
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}
A001055(n) = fcnt(n, n) \\ This function from Michael B. Porter, Oct 29 2009
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Leroy Quet, Oct 04 2000
EXTENSIONS
More terms from James A. Sellers, Oct 09 2000
More terms from Vladeta Jovovic, Nov 19 2000
STATUS
approved