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A239312
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Number of condensed integer partitions of n.
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53
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1, 1, 1, 2, 3, 3, 5, 6, 9, 10, 14, 16, 23, 27, 33, 41, 51, 62, 75, 93, 111, 134, 159, 189, 226, 271, 317, 376, 445, 520, 609, 714, 832, 972, 1129, 1304, 1520, 1753, 2023, 2326, 2692, 3077, 3540, 4050, 4642, 5298, 6054, 6887, 7854, 8926, 10133, 11501, 13044
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OFFSET
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0,4
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COMMENTS
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Suppose that p is a partition of n. Let x(1), x(2), ..., x(k) be the distinct parts of p, and let m(i) be the multiplicity of x(i) in p. Let c(p) be the partition {m(1)*x(1), m(2)*x(2), ..., x(k)*m(k)} of n. Call a partition q of n a condensed partition of n if q = c(p) for some partition p of n. Then a(n) is the number of distinct condensed partitions of n. Note that c(p) = p if and only if p has distinct parts and that condensed partitions can have repeated parts.
Also the number of integer partitions of n such that it is possible to choose a different divisor of each part. For example, the partition (6,4,4,1) has choices (3,2,4,1), (3,4,2,1), (6,2,4,1), (6,4,2,1) so is counted under a(15). - Gus Wiseman, Mar 12 2024
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LINKS
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EXAMPLE
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a(5) = 3 gives the number of partitions of 5 that result from condensations as shown here: 5 -> 5, 41 -> 41, 32 -> 32, 311 -> 32, 221 -> 41, 2111 -> 32, 11111 -> 5.
The a(1) = 1 through a(9) = 10 condensed partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(2,1) (2,2) (3,2) (3,3) (4,3) (4,4) (5,4)
(3,1) (4,1) (4,2) (5,2) (5,3) (6,3)
(5,1) (6,1) (6,2) (7,2)
(3,2,1) (3,2,2) (7,1) (8,1)
(4,2,1) (3,3,2) (4,3,2)
(4,2,2) (4,4,1)
(4,3,1) (5,2,2)
(5,2,1) (5,3,1)
(6,2,1)
(End)
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, {[]},
`if`(i=1, {[n]}, {seq(map(x-> `if`(j=0, x,
sort([x[], i*j])), b(n-i*j, i-1))[], j=0..n/i)}))
end:
a:= n-> nops(b(n$2)):
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MATHEMATICA
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u[n_, k_] := u[n, k] = Map[Total, Split[IntegerPartitions[n][[k]]]]; t[n_] := t[n] = DeleteDuplicates[Table[Sort[u[n, k]], {k, 1, PartitionsP[n]}]]; Table[Length[t[n]], {n, 0, 30}]
Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#], UnsameQ@@#&]]>0&]], {n, 0, 30}] (* Gus Wiseman, Mar 12 2024 *)
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CROSSREFS
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The complement is counted by A370320.
The version for prime factors (not all divisors) is A370592, ranks A368100.
A237685 counts partitions of depth 1, or A353837 if we include depth 0.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A355535, A355733, A355739, A367867, A368097, A368414, A370583, A370584, A370594, A370806, A370808.
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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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