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Inverse of the Hardy-Ramanujan asymptotic partition function.
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%I #15 Mar 04 2019 18:45:11

%S 1,2,3,3,4,4,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,

%T 9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,

%U 11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12

%N Inverse of the Hardy-Ramanujan asymptotic partition function.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_function_(number_theory)">Partition function</a>

%F a(n) = 6*LambertW(-1, -Pi/(2*sqrt(2)*3^(3/4)*sqrt(n)))^2/Pi^2 rounded to the nearest integer.

%F Conjecture: a(A000041(n)) = n for all n > 9.

%e A000041(10) = 42, then a(42) = 10.

%t a[n_] := 6*ProductLog[-1, -Pi/(2*Sqrt[2]*3^(3/4)*Sqrt[n])]^2/Pi^2 // Round;

%t Table[a[n], {n, 2, 100}]

%Y Cf. A000041, A050811.

%K nonn

%O 2,2

%A _Jean-François Alcover_, Mar 02 2019