A379736 Number of integer partitions of n whose product of parts is not n.

1, 0, 1, 2, 3, 6, 9, 14, 19, 28, 40, 55, 73, 100, 133, 174, 226, 296, 381, 489, 623, 790, 1000, 1254, 1568, 1956, 2434, 3007, 3714, 4564, 5599, 6841, 8342, 10141, 12308, 14881, 17968, 21636, 26013, 31183, 37331, 44582, 53169, 63260, 75171, 89130, 105556

Offset

0,4

Comments

These partitions are ranked by A379722, complement A301987.

Links

Table of n, a(n) for n=0..46.

Formula

a(n) = A000041(n) - A001055(n).

Example

The a(2) = 1 through a(7) = 14 partitions:
  (11)  (21)   (31)    (32)     (33)      (43)
        (111)  (211)   (41)     (42)      (52)
               (1111)  (221)    (51)      (61)
                       (311)    (222)     (322)
                       (2111)   (411)     (331)
                       (11111)  (2211)    (421)
                                (3111)    (511)
                                (21111)   (2221)
                                (111111)  (3211)
                                          (4111)
                                          (22111)
                                          (31111)
                                          (211111)
                                          (1111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Times@@#!=n&]], {n, 0, 30}]

Crossrefs

The complement is counted by A001055.

The strict case is A111133 (except first term).

A000041 counts integer partitions, strict A000009.

A002865 counts partitions into parts > 1, see A379734, strict A379735.

A324851 finds numbers > 1 divisible by the sum of their prime indices.

A379666 counts partitions by sum and product, without 1's A379668.

Counting and ranking multisets by comparing sum and product:

- same: A001055, ranks A301987

- divisible: A057567, ranks A326155

- divisor: A057568, ranks A326149, see A379733

- greater than: A096276 shifted right, ranks A325038

- greater or equal: A096276, ranks A325044

- less than: A114324, ranks A325037, see A318029, A379720

- less or equal: A319005, ranks A379721, see A025147

- different: A379736 (this), ranks A379722

Cf. A028422, A069016, A319000, A319916, A325036, A325041, A326152, A379671, A379678.

Sequence in context: A261243 A061925 A073736 * A101593 A349502 A226893

Adjacent sequences: A379733 A379734 A379735 * A379737 A379738 A379739

Keyword

nonn

Author

Gus Wiseman, Jan 07 2025

Status

approved