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A130689 Number of partitions of n such that every part divides the largest part; a(0) = 1. 25
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 (list; graph; refs; listen; history; text; internal format)
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
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.
Sequence in context: A003238 A051839 A130714 * A024560 A000039 A302600
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Jul 01 2007
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)