A130714 Number of partitions of n such that every part divides the largest part and such that the smallest part divides every part.

1, 2, 3, 5, 6, 10, 11, 16, 19, 26, 27, 41, 42, 55, 64, 81, 83, 114, 116, 151, 168, 202, 210, 277, 289, 348, 382, 460, 478, 604, 623, 747, 812, 942, 1006, 1223, 1269, 1479, 1605, 1870, 1959, 2329, 2434, 2818, 3056, 3458, 3653, 4280, 4493, 5130, 5507, 6231, 6580

Offset: 1

Author

Vladeta Jovovic, Jul 02 2007

Keywords

• easy
• nonn

Comments

First differs from A130689 at a(11) = 27, A130689(11) = 28.

Alternative name: Number of integer partitions of n with a part divisible by and a part dividing all the other parts. With this definition we have a(0) = 1. - Gus Wiseman, Apr 18 2021

Examples

			From _Gus Wiseman_, Apr 18 2021: (Start)
The a(1) = 1 though 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)
		

Crossrefs

The second condition alone gives A083710.

The first condition alone gives A130689.

The opposite version is A343342.

The Heinz numbers of these partitions are the complement of A343343.

The half-opposite versions are A343344 and A343345.

The complement is counted by A343346.

The strict case is A343378.

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.

Cf. A338470, A341450, A342193, A343337, A343338, A343341, A343379, A343382.

Programs

Maple

A130714 := proc(n) local gf,den,i,k,j ; gf := 0 ; for i from 0 to n do for j from 1 to n/(1+i) do den := 1 ; for k in numtheory[divisors](i) do den := den*(1-x^(j*k)) ; od ; gf := taylor(gf+x^(j+i*j)/den,x=0,n+1) ; od ; od: coeftayl(gf,x=0,n) ; end: seq(A130714(n),n=1..60) ; # R. J. Mathar, Oct 28 2007

Mathematica

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

Formula

G.f.: Sum_{i>=0} Sum_{j>0} x^(j+i*j)/Product_{k|i} (1-x^(j*k)).

Extensions

More terms from R. J. Mathar, Oct 28 2007