|
%I #45 May 11 2023 09:20:07
%S 2,3,4,6,9,13,18,26,35,48,65,87,115,152,199,258,333,427,545,692,875,
%T 1102,1381,1725,2145,2659,3285,4046,4967,6080,7423,9037,10974,13293,
%U 16065,19370,23304,27977,33519,40080,47833,56981,67757,80431,95316
%N Partition numbers rounded to nearest integer given by the Hardy-Ramanujan approximate formula.
%C The mounting error seems to be approximately A035949(n-3), n >= 4. - _Alonso del Arte_, Jul 28 2011
%C This conjecture is false, for correct approximation see the formula below. - _Vaclav Kotesovec_, Apr 03 2017
%D John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 95.
%H Dr. Math, <a href="http://mathforum.org/dr.math/problems/partitions.html">Partitioning the Integers</a>
%H Dr. Math, <a href="http://mathforum.org/dr.math/problems/huckin11.14.98.html">Partitioning an Integer</a>
%H D. Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/95/partitions">Additive Partitions of Number</a>
%H F. Ruskey, <a href="http://combos.org/part">Generate Numerical Partitions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>
%H OEIS Wiki, <a href="/wiki/Partition_function">Partition function</a>
%F a(n) = round(exp(Pi*sqrt(2*n/3))/(4*n*sqrt(3))). - _Alonso del Arte_, May 21 2011
%F 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
%p 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
%t 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 *)
%o (UBASIC) input N:print round(#e^(pi(1)*sqrt(2*N/3))/(4*N*sqrt(3)))
%o (PARI) a(n)=round(exp(Pi*sqrt(2*n/3))/(4*n*sqrt(3))) \\ _Charles R Greathouse IV_, May 01 2012
%Y Cf. A000041, A035949, A049575, A051143.
%K nonn,easy
%O 1,1
%A _Patrick De Geest_, Oct 15 1999
%E a(1) = 1 replaced by 2, a(2) = 2 replaced by 3. - _Alonso del Arte_, _D. S. McNeil_, Aug 07 2011
|
