A370805 Number of condensed integer partitions of n into parts > 1.

1, 0, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 15, 18, 22, 27, 34, 41, 51, 62, 75, 90, 109, 129, 153, 185, 217, 258, 307, 359, 421

Offset

0,5

Comments

These are partitions without ones such that it is possible to choose a different divisor of each part.

Links

Table of n, a(n) for n=0..30.

Example

The a(0) = 1 through a(9) = 6 partitions:
  ()  .  (2)  (3)  (4)    (5)    (6)    (7)      (8)      (9)
                   (2,2)  (3,2)  (3,3)  (4,3)    (4,4)    (5,4)
                                 (4,2)  (5,2)    (5,3)    (6,3)
                                        (3,2,2)  (6,2)    (7,2)
                                                 (3,3,2)  (4,3,2)
                                                 (4,2,2)  (5,2,2)

Mathematica

Table[Length[Select[IntegerPartitions[n], FreeQ[#, 1] && Length[Select[Tuples[Divisors/@#], UnsameQ@@#&]]>0&]], {n, 0, 30}]

Crossrefs

The version with ones is A239312, complement A370320.

These partitions have as ranks the odd terms of A368110, complement A355740.

The version for prime factors is A370592, complement A370593, post A370807.

The complement without ones is A370804, ranked by the odd terms of A355740.

The version for factorizations is A370814, complement A370813.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

A355731 counts choices of a divisor of each prime index, firsts A355732.

Cf. A355529, A355739, A367867, A367901, A368110, A368413, A370595, A370806, A370808, A370810.

Sequence in context: A008806 A356607 A366843 * A238860 A144367 A370778

Adjacent sequences: A370802 A370803 A370804 * A370806 A370807 A370808

Keyword

nonn,more

Author

Gus Wiseman, Mar 04 2024

Status

approved