A050811 Partition numbers rounded to nearest integer given by the Hardy-Ramanujan approximate formula.

2, 3, 4, 6, 9, 13, 18, 26, 35, 48, 65, 87, 115, 152, 199, 258, 333, 427, 545, 692, 875, 1102, 1381, 1725, 2145, 2659, 3285, 4046, 4967, 6080, 7423, 9037, 10974, 13293, 16065, 19370, 23304, 27977, 33519, 40080, 47833, 56981, 67757, 80431, 95316

Offset: 1

Author

Patrick De Geest, Oct 15 1999

Keywords

• nonn
• easy

Comments

The mounting error seems to be approximately A035949(n-3), n >= 4. - Alonso del Arte, Jul 28 2011

This conjecture is false, for correct approximation see the formula below. - Vaclav Kotesovec, Apr 03 2017

References

• John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 95.

Links

• Dr. Math, Partitioning the Integers
• Dr. Math, Partitioning an Integer
• D. Rusin, Additive Partitions of Number
• F. Ruskey, Generate Numerical Partitions
• Eric Weisstein's World of Mathematics, Partition Function P
• OEIS Wiki, Partition function

Crossrefs

Cf. A000041, A035949, A049575, A051143.

Programs

Maple

A050811:=n->round(exp(Pi*sqrt(2*n/3))/(4*n*sqrt(3))): seq(A050811(n), n=1..70); # Wesley Ivan Hurt, Sep 11 2015

Mathematica

f[n_] := Round[ E^(Sqrt[2n/3] Pi)/(4Sqrt[3] n)]; Array[f, 45] (* Alonso del Arte, May 21 2011, corrected by Robert G. Wilson v, Sep 11 2015 *)

PARI

a(n)=round(exp(Pi*sqrt(2*n/3))/(4*n*sqrt(3))) \\ Charles R Greathouse IV, May 01 2012

UBASIC

input N:print round(#e^(pi(1)*sqrt(2*N/3))/(4*N*sqrt(3)))

Formula

a(n) = round(exp(Pi*sqrt(2*n/3))/(4*n*sqrt(3))). - Alonso del Arte, May 21 2011

a(n) - A000041(n) ~ (1/Pi + Pi/72) * exp(sqrt(2*n/3)*Pi) / (4*sqrt(2)*n^(3/2)) * (1 - (9 + Pi^2/48)*Pi/((72 + Pi^2)*sqrt(6*n))). - Vaclav Kotesovec, Apr 03 2017

Extensions

a(1) = 1 replaced by 2, a(2) = 2 replaced by 3. - Alonso del Arte, D. S. McNeil, Aug 07 2011