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

1,1

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.

Links

Table of n, a(n) for n=1..40.

Eric Weisstein, Stirling Series, MathWorld.

Wikipedia, Hypersphere

Formula

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

Example

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.

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):

Crossrefs

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

Sequence in context: A005173 A196144 A294682 * A045822 A065595 A267472

Adjacent sequences: A177674 A177675 A177676 * A177678 A177679 A177680

Keyword

nonn

Author

Michel Lagneau, May 10 2010

Extensions

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

Status

approved