|
|
A130714
|
|
Number of partitions of n such that every part divides the largest part and such that the smallest part divides every part.
|
|
15
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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
|
|
LINKS
|
|
|
FORMULA
|
G.f.: Sum_{i>=0} Sum_(j>0} x^(j+i*j)/Product_{k|i} (1-x^(j*k)).
|
|
EXAMPLE
|
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)
|
|
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 *)
|
|
CROSSREFS
|
The second condition alone gives A083710.
The first condition alone gives A130689.
The Heinz numbers of these partitions are the complement of A343343.
The complement is counted by A343346.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|