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A325271
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Number of integer partitions of n with frequency depth round(sqrt(n)).
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2
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1, 1, 1, 1, 2, 1, 3, 4, 6, 8, 11, 11, 19, 44, 53, 63, 83, 113, 124, 171, 190, 344, 429, 525, 662, 796, 981, 1182, 1442, 1709, 2096, 2663, 3406, 4315, 5426, 6784, 8417, 10466, 12824, 15721, 19104, 23267, 27981, 5, 14, 36, 76, 143, 269, 446, 738, 1143, 1754
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OFFSET
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0,5
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COMMENTS
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The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).
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LINKS
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EXAMPLE
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The a(2) = 1 through a(10) = 11 partitions:
(2) (111) (22) (11111) (33) (43) (53) (54) (64)
(1111) (222) (52) (62) (63) (73)
(111111) (61) (71) (72) (82)
(421) (431) (81) (91)
(521) (432) (532)
(3311) (531) (541)
(621) (631)
(222111) (721)
(3322)
(4321)
(4411)
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MATHEMATICA
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fdadj[ptn_List]:=If[ptn=={}, 0, Length[NestWhileList[Sort[Length/@Split[#]]&, ptn, Length[#]>1&]]];
Table[Length[Select[IntegerPartitions[n], fdadj[#]==Round[Sqrt[n]]&]], {n, 0, 30}]
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CROSSREFS
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Cf. A117571, A181819, A225485, A323014, A323023, A325245, A325246, A325272, A325253, A325258, A325278, A325280, A325282.
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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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