%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