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A340693
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Number of integer partitions of n where each part is a divisor of the number of parts.
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7
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1, 1, 1, 2, 2, 3, 2, 5, 5, 7, 7, 10, 10, 14, 14, 17, 19, 24, 24, 32, 33, 42, 43, 58, 59, 75, 79, 98, 104, 124, 128, 156, 166, 196, 204, 239, 251, 292, 306, 352, 372, 426, 445, 514, 543, 616, 652, 745, 790, 896, 960, 1080, 1162, 1311, 1400, 1574, 1692, 1892
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OFFSET
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0,4
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COMMENTS
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The only strict partitions counted are (), (1), and (2,1).
Is there a simple generating function?
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LINKS
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EXAMPLE
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The a(1) = 1 through a(9) = 7 partitions:
1 11 21 22 311 2211 331 2222 333
111 1111 2111 111111 2221 4211 4221
11111 4111 221111 51111
211111 311111 222111
1111111 11111111 321111
21111111
111111111
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], And@@IntegerQ/@(Length[#]/#)&]], {n, 0, 30}]
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CROSSREFS
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Note: Heinz numbers are given in parentheses below.
The Heinz numbers of these partitions are A340606.
The version for factorizations is A340851, with reciprocal version A340853.
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A326843 = partitions of n whose length and maximum both divide n (A326837).
A330950 = partitions of n whose Heinz number is divisible by n (A324851).
Cf. A000041, A003114, A006141, A033630, A064174, A074761, A102627, A200750, A237984, A298423, A340827.
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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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