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A130714
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Number of partitions of n such that every part divides the largest part and such that the smallest part divides every part.
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15
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1, 2, 3, 5, 6, 10, 11, 16, 19, 26, 27, 41, 42, 55, 64, 81, 83, 114, 116, 151, 168, 202, 210, 277, 289, 348, 382, 460, 478, 604, 623, 747, 812, 942, 1006, 1223, 1269, 1479, 1605, 1870, 1959, 2329, 2434, 2818, 3056, 3458, 3653, 4280, 4493, 5130, 5507, 6231, 6580
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OFFSET
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1,2
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COMMENTS
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First differs from A130689 at a(11) = 27, A130689(11) = 28.
Alternative name: Number of integer partitions of n with a part divisible by and a part dividing all the other parts. With this definition we have a(0) = 1. - Gus Wiseman, Apr 18 2021
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LINKS
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Table of n, a(n) for n=1..53.
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FORMULA
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G.f.: Sum_{i>=0} Sum_(j>0} x^(j+i*j)/Product_{k|i} (1-x^(j*k)).
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EXAMPLE
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From Gus Wiseman, Apr 18 2021: (Start)
The a(1) = 1 though 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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MAPLE
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A130714 := proc(n) local gf, den, i, k, j ; gf := 0 ; for i from 0 to n do for j from 1 to n/(1+i) do den := 1 ; for k in numtheory[divisors](i) do den := den*(1-x^(j*k)) ; od ; gf := taylor(gf+x^(j+i*j)/den, x=0, n+1) ; od ; od: coeftayl(gf, x=0, n) ; end: seq(A130714(n), n=1..60) ; # R. J. Mathar, Oct 28 2007
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MATHEMATICA
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Table[If[n==0, 1, Length[Select[IntegerPartitions[n], And@@IntegerQ/@(#/Min@@#)&&And@@IntegerQ/@(Max@@#/#)&]]], {n, 0, 30}] (* Gus Wiseman, Apr 18 2021 *)
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CROSSREFS
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The second condition alone gives A083710.
The first condition alone gives A130689.
The opposite version is A343342.
The Heinz numbers of these partitions are the complement of A343343.
The half-opposite versions are A343344 and A343345.
The complement is counted by A343346.
The strict case is A343378.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
Cf. A338470, A341450, A342193, A343337, A343338, A343341, A343379, A343382.
Sequence in context: A018396 A003238 A051839 * A130689 A024560 A000039
Adjacent sequences: A130711 A130712 A130713 * A130715 A130716 A130717
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KEYWORD
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easy,nonn
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AUTHOR
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Vladeta Jovovic, Jul 02 2007
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EXTENSIONS
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More terms from R. J. Mathar, Oct 28 2007
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STATUS
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approved
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