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%I #10 Apr 22 2021 01:42:23
%S 1,0,1,1,2,1,4,1,5,3,7,2,13,2,13,9,17,6,27,7,33,19,35,16,58,22,58,39,
%T 75,37,108,44,117,75,132,88,190,94,199,147,250,153,322,180,363,271,
%U 405,286,544,339,601,458,699,503,868,608,990,777,1113,865,1422,993
%N Number of integer partitions of n with no 1's and a part divisible by all the other parts.
%C Alternative name: Number of integer partitions of n with no 1's that are empty or have greatest part divisible by all the other parts.
%e The a(6) = 4 through a(16) = 17 partitions (A..G = 10..16):
%e 6 7 8 9 A B C D E F G
%e 33 44 63 55 632 66 6322 77 A5 88
%e 42 62 333 82 84 C2 C3 C4
%e 222 422 442 93 662 555 E2
%e 2222 622 A2 842 663 844
%e 4222 444 A22 933 C22
%e 22222 633 4442 6333 4444
%e 822 6332 33333 6622
%e 3333 8222 63222 8422
%e 4422 44222 A222
%e 6222 62222 44422
%e 42222 422222 63322
%e 222222 2222222 82222
%e 442222
%e 622222
%e 4222222
%e 22222222
%t Table[If[n==0,1,Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Or@@And@@IntegerQ/@(Max@@#/#)&]]],{n,0,30}]
%Y The dual version is A083711.
%Y The version with 1's allowed is A130689.
%Y The strict case is A339660.
%Y The Heinz numbers of these partitions are the odd complement of A343337.
%Y The strict case with 1's allowed is A343347.
%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 Cf. A066186, A083710, A083711, A097986, A130714, A338470, A343341, A343342, A343346, A343377, A343382.
%K nonn
%O 0,5
%A _Gus Wiseman_, Apr 18 2021
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