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

1,1

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

Table of n, a(n) for n=1..45.

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

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

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 *)

Prog

(UBASIC) input N:print round(#e^(pi(1)*sqrt(2*N/3))/(4*N*sqrt(3)))
(PARI) a(n)=round(exp(Pi*sqrt(2*n/3))/(4*n*sqrt(3))) \\ Charles R Greathouse IV, May 01 2012

Crossrefs

Cf. A000041, A035949, A049575, A051143.

Sequence in context: A219282 A098578 A303667 * A076968 A238430 A285484

Adjacent sequences: A050808 A050809 A050810 * A050812 A050813 A050814

Keyword

nonn,easy

Author

Patrick De Geest, Oct 15 1999

Extensions

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

Status

approved