A177677 The maximum integer dimension in which the volume of the hypersphere of radius n remains larger than 1.

12, 62, 147, 266, 419, 607, 828, 1084, 1375, 1699, 2057, 2450, 2877, 3338, 3833, 4362, 4926, 5523, 6155, 6821, 7521, 8256, 9024, 9827, 10664, 11535, 12440, 13379, 14353, 15360, 16402, 17478, 18588, 19732, 20911, 22123, 23370, 24651, 25966, 27315

Offset: 1

Author

Michel Lagneau, May 10 2010

Keywords

• nonn

Comments

The volume of the d-dimensional hypersphere of radius n is V= Pi^(d/2) * n^d / Gamma(1 + d/2).

For fixed radius, V -> 0 as d->infinity, so there is a dimension d for which V(n,d) > 1 but V(n,d+1) < 1, which defines the entry in the sequence.

Examples

			a(n=2)=62 because Pi^(62/2) * 2^62/GAMMA(1 + (62/2)) =1.447051 and Pi^(63/2)* 2^63 / Gamma(1 + (63/2)) =0.9103541.
		

Links

• Eric Weisstein, Stirling Series, MathWorld.
• Wikipedia, Hypersphere

Crossrefs

Cf. A005446 , A005147, A001164, A005146, A005447, A001163

Programs

Maple

with(numtheory): n0:=50: T:=array(1..n0): for r from 1 to n0 do: x:=2: for n from 1 to 1000000 while(x>=1) do: x:= floor(evalf((r^n * Pi^(n/2))/GAMMA(1 + n/2))):k:=n:od:T[r]:=k-1:od:print(T):

Formula

a(n) = max {d: Pi^d/2 * n^d / Gamma(1+d/2) > 1}.

Extensions

Use of variables standardized. Definition simplified, comments tightened, unspecific reference and superfluous parentheses removed - R. J. Mathar, Oct 20 2010