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A050811
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Partition numbers rounded to nearest integer given by the Hardy-Ramanujan approximate formula.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 95.
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LINKS
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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
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000041, A035949, A049575, A051143.
Sequence in context: A219282 A098578 A303667 * A076968 A238430 A285484
Adjacent sequences: A050808 A050809 A050810 * A050812 A050813 A050814
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KEYWORD
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nonn,easy
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AUTHOR
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Patrick De Geest, Oct 15 1999
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EXTENSIONS
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a(1) = 1 replaced by 2, a(2) = 2 replaced by 3. - Alonso del Arte, D. S. McNeil, Aug 07 2011
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STATUS
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approved
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