A387327 Number of ways to choose an integer partition of each prime factor of n (with multiplicity).

1, 2, 3, 4, 7, 6, 15, 8, 9, 14, 56, 12, 101, 30, 21, 16, 297, 18, 490, 28, 45, 112, 1255, 24, 49, 202, 27, 60, 4565, 42, 6842, 32, 168, 594, 105, 36, 21637, 980, 303, 56, 44583, 90, 63261, 224, 63, 2510, 124754, 48, 225, 98, 891, 404, 329931, 54, 392, 120

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Formula

Totally multiplicative with a(p) = A000041(p) for prime p. - Andrew Howroyd, Nov 10 2025

Sum_{n>=1} 1/a(n) = 1/Product_{n>=1} (1-1/A058698(n)) = 3.88219666056645519287... . - Amiram Eldar, Nov 12 2025

Example

The a(1) = 1 through a(7) = 15 ways:
  (1)  (2)   (3)    (2)(2)    (5)      (2)(3)     (7)
       (11)  (21)   (11)(2)   (32)     (11)(3)    (43)
             (111)  (2)(11)   (41)     (2)(21)    (52)
                    (11)(11)  (221)    (11)(21)   (61)
                              (311)    (2)(111)   (322)
                              (2111)   (11)(111)  (331)
                              (11111)             (421)
                                                  (511)
                                                  (2221)
                                                  (3211)
                                                  (4111)
                                                  (22111)
                                                  (31111)
                                                  (211111)
                                                  (1111111)

Maple

a:= n-> mul(combinat[numbpart](i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..56);  # Alois P. Heinz, Nov 10 2025

Mathematica

Table[Length[Tuples[IntegerPartitions/@Flatten[ConstantArray@@@FactorInteger[n]]]], {n, 30}]

Prog

(PARI) a(n) = { my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); numbpart(p)^e) } \\ Andrew Howroyd, Nov 10 2025

Crossrefs

For constant partitions we have A061142, for prime indices A355731.

For prime indices instead of factors we have A299200.

The version for distinct choices is A387133, zeros A387326.

A000041 counts integer partitions, strict A000009.

A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.

A387110 counts choices of distinct distinct integer partitions of each prime index.

Cf. A058698, A063834, A120383, A299201, A335433, A335448, A355741, A357977, A383706, A387115, A387120, A387136.

Sequence in context: A064554 A290641 A340069 * A265352 A265368 A239972

Adjacent sequences: A387324 A387325 A387326 * A387328 A387329 A387330

Keyword

nonn,mult

Author

Gus Wiseman, Sep 05 2025

Status

approved