This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A239312 #30 Mar 13 2024 19:20:10
%S A239312 1,1,1,2,3,3,5,6,9,10,14,16,23,27,33,41,51,62,75,93,111,134,159,189,
%T A239312 226,271,317,376,445,520,609,714,832,972,1129,1304,1520,1753,2023,
%U A239312 2326,2692,3077,3540,4050,4642,5298,6054,6887,7854,8926,10133,11501,13044
%N A239312 Number of condensed integer partitions of n.
%C A239312 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.
%C A239312 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
%H A239312 Alois P. Heinz, <a href="/A239312/b239312.txt">Table of n, a(n) for n = 0..100</a> (first 84 terms from Manfred Scheucher)
%H A239312 Manfred Scheucher, <a href="/A239312/a239312.py.txt">Python Script</a>
%e A239312 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.
%e A239312 From _Gus Wiseman_, Mar 12 2024: (Start)
%e A239312 The a(1) = 1 through a(9) = 10 condensed partitions:
%e A239312 (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e A239312 (2,1) (2,2) (3,2) (3,3) (4,3) (4,4) (5,4)
%e A239312 (3,1) (4,1) (4,2) (5,2) (5,3) (6,3)
%e A239312 (5,1) (6,1) (6,2) (7,2)
%e A239312 (3,2,1) (3,2,2) (7,1) (8,1)
%e A239312 (4,2,1) (3,3,2) (4,3,2)
%e A239312 (4,2,2) (4,4,1)
%e A239312 (4,3,1) (5,2,2)
%e A239312 (5,2,1) (5,3,1)
%e A239312 (6,2,1)
%e A239312 (End)
%p A239312 b:= proc(n,i) option remember; `if`(n=0, {[]},
%p A239312 `if`(i=1, {[n]}, {seq(map(x-> `if`(j=0, x,
%p A239312 sort([x[], i*j])), b(n-i*j, i-1))[], j=0..n/i)}))
%p A239312 end:
%p A239312 a:= n-> nops(b(n$2)):
%p A239312 seq(a(n), n=0..50); # _Alois P. Heinz_, Jul 01 2019
%t A239312 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}]
%t A239312 Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#],UnsameQ@@#&]]>0&]], {n,0,30}] (* _Gus Wiseman_, Mar 12 2024 *)
%Y A239312 The strict case is A000009.
%Y A239312 These partitions have ranks A368110, complement A355740.
%Y A239312 The complement is counted by A370320.
%Y A239312 The version for prime factors (not all divisors) is A370592, ranks A368100.
%Y A239312 The complement for prime factors is A370593, ranks A355529.
%Y A239312 For a unique choice we have A370595, ranks A370810.
%Y A239312 For multiple choices we have A370803, ranks A370811.
%Y A239312 The case without ones is A370805, complement A370804.
%Y A239312 The version for factorizations is A370814, complement A370813.
%Y A239312 A000005 counts divisors.
%Y A239312 A000041 counts integer partitions.
%Y A239312 A237685 counts partitions of depth 1, or A353837 if we include depth 0.
%Y A239312 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A239312 Cf. A355535, A355733, A355739, A367867, A368097, A368414, A370583, A370584, A370594, A370806, A370808.
%K A239312 nonn
%O A239312 0,4
%A A239312 _Clark Kimberling_, Mar 15 2014
%E A239312 Typo in definition corrected by _Manfred Scheucher_, May 29 2015
%E A239312 Name edited by _Gus Wiseman_, Mar 13 2024