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%I #12 Jan 28 2022 01:28:03
%S 0,1,1,2,1,2,2,3,2,2,3,3,3,3,3,4,3,3,4,3,4,4,4,4,4,4,4,4,4,4,4,5,4,4,
%T 5,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,5,5,6,5,
%U 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6
%N Number of condensations to convert all the partitions of n to strict partitions of n.
%C 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. The partition {m(1)*x(1), m(2)*x(2), ..., x(k)*m(k)} of n is called the condensation of p.
%H Rémy Sigrist, <a href="/A326370/a326370.gp.txt">PARI program for A326370</a>
%e The condensation of [4, 2, 1, 1] is [4, 2, 2], of which the condensation is [4, 4], of which condensation is [8]; thus, a total of three condensations. This is maximal for the partitions of 8, so that a(8) = 3. See A239312.
%t f[m_] := Table[Tally[m][[h]][[1]]*Tally[m][[h]][[2]], {h, 1, Length[Tally[m]]}];
%t m[n_, k_] := IntegerPartitions[n][[k]];
%t q[n_, k_] := -2 + Length[FixedPointList[f, m[n, k]]];
%t a[n_] := Max[Table[q[n, k], {k, 1, PartitionsP[n]}]];
%t Table[a[n], {n, 1, 30}]
%o (PARI) See Links section.
%Y Cf. A000009, A000041, A239312, A326371.
%K nonn,easy
%O 1,4
%A _Clark Kimberling_, Jul 06 2019
%E More terms from _Rémy Sigrist_, Jul 07 2019
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