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A177677 The maximum integer dimension in which the volume of the hypersphere of radius n remains larger than 1. 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 7 16:00 EDT 2021. Contains 343652 sequences. (Running on oeis4.)