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A167932
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Number of partitions of n such that all parts are equal or all parts are distinct.
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3
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1, 1, 2, 3, 4, 4, 7, 6, 9, 10, 13, 13, 20, 19, 25, 30, 36, 39, 51, 55, 69, 79, 92, 105, 129, 144, 168, 195, 227, 257, 303, 341, 395, 451, 515, 588, 676, 761, 867, 985, 1120, 1261, 1433, 1611, 1821, 2053, 2307, 2591, 2919, 3266, 3663, 4100, 4587, 5121, 5725, 6381
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OFFSET
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0,3
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COMMENTS
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Note that for positive integers the number of partitions of n such that all parts are equal is equal to the number of proper divisors of n. (A032741(n)).
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LINKS
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FORMULA
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EXAMPLE
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The partitions of 6 are:
6 .............. All parts are distinct ..... (1).
5+1 ............ All parts are distinct ..... (2).
4+2 ............ All parts are distinct ..... (3).
4+1+1 .......... Only some parts are equal.
3+3 ............ All parts are equal ........ (4).
3+2+1 .......... All parts are distinct ..... (5).
3+1+1+1 ........ Only some parts are equal.
2+2+2 .......... All parts are equal ........ (6).
2+2+1+1 ........ Only some parts are equal.
2+1+1+1+1 ...... Only some parts are equal.
1+1+1+1+1+1 .... All parts are equal ........ (7).
So a(6) = 7.
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MATHEMATICA
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ds[n_]:=Module[{lun=Length[Union[n]]}, Length[n]==lun||lun==1]; Table[ Count[ IntegerPartitions[n], _?(ds)], {n, 0, 60}] (* Harvey P. Dale, Sep 13 2011 *)
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CROSSREFS
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Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A134400, A135010, A138121, A167930, A167931, A167933.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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