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A130689 Number of partitions of n such that every part divides the largest part; a(0) = 1. 25

%I #25 Apr 26 2021 21:26:45

%S 1,1,2,3,5,6,10,11,16,19,26,28,41,43,56,65,82,88,115,122,155,174,209,

%T 225,283,305,363,402,477,514,622,666,783,858,990,1078,1268,1362,1561,

%U 1708,1958,2111,2433,2613,2976,3247,3652,3938,4482,4821,5422

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

%C First differs from A130714 at a(11) = 28, A130714(11) = 27. - _Gus Wiseman_, Apr 23 2021

%H Alois P. Heinz, <a href="/A130689/b130689.txt">Table of n, a(n) for n = 0..5000</a> (first 1001 terms from Andrew Howroyd)

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

%e 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].

%e From _Gus Wiseman_, Apr 18 2021: (Start)

%e The a(1) = 1 through a(8) = 16 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (21) (22) (41) (33) (61) (44)

%e (111) (31) (221) (42) (331) (62)

%e (211) (311) (51) (421) (71)

%e (1111) (2111) (222) (511) (422)

%e (11111) (411) (2221) (611)

%e (2211) (4111) (2222)

%e (3111) (22111) (3311)

%e (21111) (31111) (4211)

%e (111111) (211111) (5111)

%e (1111111) (22211)

%e (41111)

%e (221111)

%e (311111)

%e (2111111)

%e (11111111)

%e (End)

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

%o (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

%Y Cf. A018818, A117086.

%Y The dual version is A083710.

%Y The case without 1's is A339619.

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

%Y The complement is counted by A343341.

%Y The strict case is A343347.

%Y The complement in the strict case is counted by A343377.

%Y A000009 counts strict partitions.

%Y A000041 counts partitions.

%Y A000070 counts partitions with a selected part.

%Y A006128 counts partitions with a selected position.

%Y A015723 counts strict partitions with a selected part.

%Y A072233 counts partitions by sum and greatest part.

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

%K easy,nonn

%O 0,3

%A _Vladeta Jovovic_, Jul 01 2007

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