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, 493, 577, 675, 788, 909, 1062, 1227, 1418, 1633, 1894, 2169, 2497, 2860, 3285, 3754, 4298, 4894, 5587, 6359, 7230, 8215, 9331, 10567, 11965

Offset: 0

Author

Gus Wiseman, Mar 04 2024

Keywords

• nonn

Comments

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

Examples

			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)
		

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.

Programs

Mathematica

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

Extensions

More terms from Jinyuan Wang, Feb 14 2025