A391228 Number of strict integer partitions of n > 0 such that the least part and the greatest part are both odd (equivalently, their product is odd).

1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 3, 7, 5, 8, 8, 12, 11, 16, 17, 23, 24, 30, 33, 42, 46, 54, 62, 75, 83, 98, 111, 130, 146, 168, 192, 222, 249, 284, 322, 368, 413, 470, 529, 599, 673, 758, 851, 961, 1074, 1204, 1348, 1511, 1685, 1884, 2100, 2344, 2606, 2902, 3225

Offset

1,6

Links

Table of n, a(n) for n=1..59.

Example

The a(8) = 3 through a(14) = 8 partitions:
  (5,3)    (9)      (7,3)    (11)       (7,5)      (13)       (9,5)
  (7,1)    (5,3,1)  (9,1)    (7,3,1)    (9,3)      (7,5,1)    (11,3)
  (5,2,1)           (5,4,1)  (5,3,2,1)  (11,1)     (9,3,1)    (13,1)
                    (7,2,1)             (5,4,3)    (5,4,3,1)  (7,4,3)
                                        (7,4,1)    (7,3,2,1)  (7,6,1)
                                        (9,2,1)               (9,4,1)
                                        (5,4,2,1)             (11,2,1)
                                                              (7,4,2,1)

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[First[#]*Last[#]]&]], {n, 30}]

Crossrefs

For odd least part we have A026832, non-strict A026804, ranks A340932.

For even least part we have A026833, non-strict A026805, ranks A340933.

For odd greatest part we have A026837, non-strict A027193, ranks A244991.

For even greatest part we have A067661, non-strict A027187, ranks A244990.

The non-strict version is A325338, ranks A390093.

For sum instead of product we have A390746, non-strict A390092, ranks A390988.

The complement is counted by A391231, non-strict A391230, ranks A391229.

For both parts even we have A391233, non-strict A325346, ranks A391232.

A000041 counts integer partitions, strict A000009.

A006141 counts partitions with least part = length, ranks A324522.

A257991 counts odd prime indices, even A257992.

A333352 gives least prime index times greatest prime index.

Cf. A005408, A035457, A058695, A066208, A067659, A101707, A160786, A300063, A300272, A340832, A391227.

Sequence in context: A008584 A352833 A034390 * A368671 A393167 A183912

Adjacent sequences: A391225 A391226 A391227 * A391229 A391230 A391231

Keyword

nonn

Author

Gus Wiseman, Dec 10 2025

Status

approved