A285482 Smallest k such that A285481(k) >= n, i.e., lowest d where the smallest integer radius needed for a d-dimensional ball to have a volume >= 1 is at least n.
1, 13, 63, 148, 267, 420, 608, 829, 1085, 1376, 1700, 2058, 2451, 2878, 3339, 3834, 4363, 4927, 5524, 6156, 6822, 7522, 8257, 9025, 9828, 10665, 11536, 12441, 13380, 14354, 15361, 16403, 17479, 18589, 19733, 20912, 22124, 23371, 24652, 25967, 27316, 28700
Offset: 1
Keywords
Examples
For n = 3: a 63-ball of radius 2 has a volume of 0.91035..., while a 63-ball of radius 3 has a volume of 112969101106.64166... Since 63 is the least number of dimensions where a ball with unit volume has a radius >= 3, a(3) = 63.
Crossrefs
Cf. A285481.
Programs
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Mathematica
a[1]=1; a[n_]:=a[n] = Block[{k = a[n-1]}, While[Ceiling[(Pi^(-k/2) Gamma[1 + k/2])^(1/k)] < n, k++]; k]; Array[a, 20] (* Giovanni Resta, Apr 29 2017 *) -
PARI
volume(n, r) = ((Pi^(n/2))/(gamma(1+n/2)))*r^n a285481(n) = my(k=1); while(1, if(volume(n, k) >= 1, return(k)); k++) a(n) = my(k=1); while(1, if(a285481(k) >= n, return(k)); k++)
Extensions
a(36)-a(42) from Giovanni Resta, Apr 29 2017