A177677 - OEIS

%I #10 Jan 30 2016 17:56:29

%S 12,62,147,266,419,607,828,1084,1375,1699,2057,2450,2877,3338,3833,

%T 4362,4926,5523,6155,6821,7521,8256,9024,9827,10664,11535,12440,13379,

%U 14353,15360,16402,17478,18588,19732,20911,22123,23370,24651,25966,27315

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

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

%C 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.

%H Eric Weisstein, <a href="http://mathworld.wolfram.com/StirlingsSeries.html">Stirling Series</a>, MathWorld.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hypersphere">Hypersphere</a>

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

%e 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.

%p 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):

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

%K nonn

%O 1,1

%A _Michel Lagneau_, May 10 2010

%E Use of variables standardized. Definition simplified, comments tightened, unspecific reference and superfluous parentheses removed - _R. J. Mathar_, Oct 20 2010