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A338470
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Number of integer partitions of n with no part dividing all the others.
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24
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1, 0, 0, 0, 0, 1, 0, 3, 2, 5, 5, 13, 7, 23, 21, 33, 35, 65, 55, 104, 97, 151, 166, 252, 235, 377, 399, 549, 591, 846, 858, 1237, 1311, 1749, 1934, 2556, 2705, 3659, 3991, 5090, 5608, 7244, 7841, 10086, 11075, 13794, 15420, 19195, 21003, 26240, 29089, 35483
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OFFSET
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0,8
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COMMENTS
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Alternative name: Number of integer partitions of n that are empty or have smallest part not dividing all the others.
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LINKS
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FORMULA
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EXAMPLE
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The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
(32) . (43) (53) (54) (64) (65) (75)
(52) (332) (72) (73) (74) (543)
(322) (432) (433) (83) (552)
(522) (532) (92) (732)
(3222) (3322) (443) (4332)
(533) (5322)
(542) (33222)
(632)
(722)
(3332)
(4322)
(5222)
(32222)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], #=={}||!And@@IntegerQ/@(#/Min@@#)&]], {n, 0, 30}]
(* Second program: *)
a[n_] := If[n == 0, 1, PartitionsP[n] - Sum[PartitionsP[d-1], {d, Divisors[n]}]];
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PROG
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(PARI) a(n)={numbpart(n) - if(n, sumdiv(n, d, numbpart(d-1)))} \\ Andrew Howroyd, Mar 25 2021
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CROSSREFS
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The Heinz numbers of these partitions are A342193.
The case with maximum part not divisible by all the others is A343342.
The case with maximum part divisible by all the others is A343344.
A000070 counts partitions with a selected part.
A001787 count normal multisets with a selected position.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
A276024 counts positive subset sums.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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