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 A239312 Number of condensed partitions of n; see Comments. 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..100 (first 84 terms from Manfred Scheucher) Manfred Scheucher, Python Script EXAMPLE 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. MAPLE 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)): seq(a(n), n=0..50);  # Alois P. Heinz, Jul 01 2019 MATHEMATICA 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}] CROSSREFS Cf. A237685. Sequence in context: A084338 A300446 A039876 * A317167 A070830 A039862 Adjacent sequences:  A239309 A239310 A239311 * A239313 A239314 A239315 KEYWORD nonn,easy AUTHOR Clark Kimberling, Mar 15 2014 EXTENSIONS Typo in definition corrected by Manfred Scheucher, May 29 2015 STATUS approved

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Last modified July 3 08:14 EDT 2020. Contains 335417 sequences. (Running on oeis4.)