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A057567 Number of partitions of n where the product of parts divides n. 23
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

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n

FORMULA

Sum_{ d divides n } A001055(d). - Vladeta Jovovic

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

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

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

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

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

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.

Sequence in context: A069932 A056148 A304442 * A217895 A328720 A005128

Adjacent sequences:  A057564 A057565 A057566 * A057568 A057569 A057570

KEYWORD

nonn

AUTHOR

Leroy Quet, Oct 04 2000

EXTENSIONS

More terms from James A. Sellers, Oct 09 2000; and from Vladeta Jovovic, Nov 19 2000

STATUS

approved

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Last modified May 30 05:35 EDT 2020. Contains 334712 sequences. (Running on oeis4.)