A306631 Inverse of the Hardy-Ramanujan asymptotic partition function.

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, 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, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12

Offset

2,2

Links

Table of n, a(n) for n=2..79.

Eric Weisstein's MathWorld, Partition Function P

Wikipedia, Partition function

Formula

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

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

Example

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

Mathematica

a[n_] := 6*ProductLog[-1, -Pi/(2*Sqrt[2]*3^(3/4)*Sqrt[n])]^2/Pi^2 // Round;
Table[a[n], {n, 2, 100}]

Crossrefs

Cf. A000041, A050811.

Sequence in context: A086592 A279783 A132663 * A023964 A328179 A000267

Adjacent sequences: A306628 A306629 A306630 * A306632 A306633 A306634

Keyword

nonn

Author

Jean-François Alcover, Mar 02 2019

Status

approved