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A130689
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Number of partitions of n such that every part divides the largest part; a(0) = 1.
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25
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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
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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G.f.: 1 + Sum_{n>0} x^n/Product_{d divides n} (1-x^d).
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EXAMPLE
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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].
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)
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MATHEMATICA
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Table[If[n==0, 1, Length[Select[IntegerPartitions[n], FreeQ[#, 1]&&And@@IntegerQ/@(Max@@#/#)&]]], {n, 0, 30}] (* Gus Wiseman, Apr 18 2021 *)
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PROG
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(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
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CROSSREFS
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The Heinz numbers of these partitions are the complement of A343337.
The complement is counted by A343341.
The complement in the strict case is counted by A343377.
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.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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