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%I #8 Jan 30 2014 11:10:18
%S 0,0,1,1,0,1,1,1,1,1,0,1,1,1,1,2,1,2,1,1,0,2,2,1,0,2,2,0,2,2,1,2,0,3,
%T 1,1,1,3,2,1,1,3,1,1,1,1,2,1,1,2,1,2,2,1,2,1,1,3,2,3,1,2,2,2,2,0,2,1,
%U 3,1,2,3,3,1,3,3,1,2,2,1,2,2,2,2,1,1,1,2,2,0,3,2,2,1,1,2,3,1,3,2,2,2,3,2,3
%N Number of primes between sqrt(n^3) and sqrt((n+1)^3).
%C The following are the only values of n such that the interval contains no primes: 0 1 4 10 20 24 27 32 65 89 121 139 141 187 207 306 321 348 1006 1051 Conjecture 1: for n>1051, a prime always exists between n^1.5 and (n+1)^1.5. Conjecture 2: for n>7295, more than 2 primes always exist between n^1.5 and (n+1)^1.5.
%e n = 2 [n^3/2] = 2 [(n+1)^3/2] = 5 there is 1 prime between 2 and 5 = 3.
%t Table[Count[Range[Floor[Surd[n^3,2]]+1,Floor[Surd[(n+1)^3,2]-1]], _?PrimeQ],{n,0,110}] (* _Harvey P. Dale_, Jan 30 2014 *)
%o (PARI) sqcubespr(n) = { for(x=0,n, ct=0; for(y=floor(sqrt(x^3))+1,floor(sqrt((x+1)^3)-1), if(isprime(y), ct++; ); ); if(ct>=0,print1(ct" ");) ) }
%K easy,nonn
%O 0,16
%A _Cino Hilliard_, Jan 05 2003
%E Comments edited by _Harvey P. Dale_, Jan 30 2014