A130689 Number of partitions of n such that every part divides the largest part; a(0) = 1.

1, 1, 2, 3, 5, 6, 10, 11, 16, 19, 26, 28, 41, 43, 56, 65, 82, 88, 115, 122, 155, 174, 209, 225, 283, 305, 363, 402, 477, 514, 622, 666, 783, 858, 990, 1078, 1268, 1362, 1561, 1708, 1958, 2111, 2433, 2613, 2976, 3247, 3652, 3938, 4482, 4821, 5422

Offset

0,3

Comments

First differs from A130714 at a(11) = 28, A130714(11) = 27. - Gus Wiseman, Apr 23 2021

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 1001 terms from Andrew Howroyd)

Formula

G.f.: 1 + Sum_{n>0} x^n/Product_{d divides n} (1-x^d).

Example

For n = 6 we have 10 such partitions: [1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 2], [1, 1, 2, 2], [2, 2, 2], [1, 1, 1, 3], [3, 3], [1, 1, 4], [2, 4], [1, 5], [6].
From Gus Wiseman, Apr 18 2021: (Start)
The a(1) = 1 through a(8) = 16 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (33)      (61)       (44)
             (111)  (31)    (221)    (42)      (331)      (62)
                    (211)   (311)    (51)      (421)      (71)
                    (1111)  (2111)   (222)     (511)      (422)
                            (11111)  (411)     (2221)     (611)
                                     (2211)    (4111)     (2222)
                                     (3111)    (22111)    (3311)
                                     (21111)   (31111)    (4211)
                                     (111111)  (211111)   (5111)
                                               (1111111)  (22211)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

Mathematica

Table[If[n==0, 1, Length[Select[IntegerPartitions[n], FreeQ[#, 1]&&And@@IntegerQ/@(Max@@#/#)&]]], {n, 0, 30}] (* Gus Wiseman, Apr 18 2021 *)

Prog

(PARI) seq(n)={Vec(1 + sum(m=1, n, my(u=divisors(m)); x^m/prod(i=1, #u, 1 - x^u[i] + O(x^(n-m+1)))))} \\ Andrew Howroyd, Apr 17 2021

Crossrefs

Cf. A018818, A117086.

The dual version is A083710.

The case without 1's is A339619.

The Heinz numbers of these partitions are the complement of A343337.

The complement is counted by A343341.

The strict case is A343347.

The complement in the strict case is counted by A343377.

A000009 counts strict partitions.

A000041 counts partitions.

A000070 counts partitions with a selected part.

A006128 counts partitions with a selected position.

A015723 counts strict partitions with a selected part.

A072233 counts partitions by sum and greatest part.

Cf. A066186, A083711, A097986, A338470, A341450, A343346, A343382.

Sequence in context: A003238 A051839 A130714 * A024560 A000039 A302600

Adjacent sequences: A130686 A130687 A130688 * A130690 A130691 A130692

Keyword

easy,nonn

Author

Vladeta Jovovic, Jul 01 2007

Status

approved