A114869 s(n) = floor(n^(n/5)/n!!!!!).

1, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 6, 5, 5, 7, 10, 16, 12, 14, 18, 26, 39, 31, 35, 45, 64, 98, 79, 88, 114, 163, 249, 200, 223, 291, 416, 636, 511, 572, 745, 1067, 1634, 1316, 1474, 1922, 2755, 4222, 3405, 3817, 4982, 7147, 10961, 8848, 9925, 12966

Offset: 1

Author

Jonathan Vos Post, Feb 20 2006

Keywords

• easy
• nonn

Comments

This sequence is an approximation of a quintuple factorial analog to Stirling's approximation to factorial. Note that a(n) is exact for n = 1, 5, 10.

Examples

			a(10) = floor(10^2/10!!!!!) = floor(10^2/50) = floor(2) = 2.
a(15) = floor(15^3/15!!!!!) = floor((15^3)/750) = floor(4.5) = 4.
a(20) = floor(20^4/20!!!!!) = floor((20^4)/15000) = floor(10.6666667) = 10.
a(25) = floor(25^5/25!!!!!) = floor((25^5)/375000) = floor(26.0416667) = 26.
a(30) = floor(30^6/30!!!!!) = floor((30^6)/11250000) = floor(64.8) = 64.
a(35) = floor(35^7/35!!!!!) = floor((35^7)/393750000) = floor(163.401389) = 163.
		

Crossrefs

Cf. A000312, A006882, A055775, A085157.

Programs

Mathematica

fac[n_Integer, m_Integer] := Block[{t = n, f = Max[1, n]}, While[t > m, t -= m; f *= t]; f]; a[n_] := Floor[n^(n/5)/fac[n, 5]]; Array[a, 65] (* Giovanni Resta, Jun 15 2016 *)

Formula

a(n) = floor(n^(n/5)/n!!!). a(n) = floor((A000312(n)^(1/5))/A085157(n)).

Extensions

Corrected and extended by Giovanni Resta, Jun 15 2016