A338470 Number of integer partitions of n with no part dividing all the others.

1, 0, 0, 0, 0, 1, 0, 3, 2, 5, 5, 13, 7, 23, 21, 33, 35, 65, 55, 104, 97, 151, 166, 252, 235, 377, 399, 549, 591, 846, 858, 1237, 1311, 1749, 1934, 2556, 2705, 3659, 3991, 5090, 5608, 7244, 7841, 10086, 11075, 13794, 15420, 19195, 21003, 26240, 29089, 35483

Offset: 0

Author

Gus Wiseman, Mar 23 2021

Keywords

• nonn

Comments

Alternative name: Number of integer partitions of n that are empty or have smallest part not dividing all the others.

Examples

			The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
  (32)  .  (43)   (53)   (54)    (64)    (65)     (75)
           (52)   (332)  (72)    (73)    (74)     (543)
           (322)         (432)   (433)   (83)     (552)
                         (522)   (532)   (92)     (732)
                         (3222)  (3322)  (443)    (4332)
                                         (533)    (5322)
                                         (542)    (33222)
                                         (632)
                                         (722)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

The complement is A083710 (strict: A097986).

The strict case is A341450.

The Heinz numbers of these partitions are A342193.

The dual version is A343341.

The case with maximum part not divisible by all the others is A343342.

The case with maximum part divisible by all the others is A343344.

A000005 counts divisors.

A000041 counts partitions.

A000070 counts partitions with a selected part.

A001787 count normal multisets with a selected position.

A006128 counts partitions with a selected position.

A015723 counts strict partitions with a selected part.

A167865 counts strict chains of divisors > 1 summing to n.

A276024 counts positive subset sums.

Sequences with similar formulas: A024994, A047966, A047968, A168111.

Cf. A001792, A064391, A064410, A066186, A067824, A083711, A098965, A264401, A339562, A339563.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]
(* Second program: *)
a[n_] := If[n == 0, 1, PartitionsP[n] - Sum[PartitionsP[d-1], {d, Divisors[n]}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 09 2021, after Andrew Howroyd *)

PARI

a(n)={numbpart(n) - if(n, sumdiv(n, d, numbpart(d-1)))} \\ Andrew Howroyd, Mar 25 2021

Formula

a(n) = A000041(n) - Sum_{d|n} A000041(d-1) for n > 0. - Andrew Howroyd, Mar 25 2021