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A291139 Primes of the form floor(k^(3/2)) for some integer k.

%I A291139 #14 Jun 02 2020 12:21:43
%S A291139 2,5,11,31,41,89,103,181,281,311,353,419,769,797,811,839,853,911,1091,
%T A291139 1153,1201,1217,1249,1499,1621,1873,2081,2557,2999,3307,3533,3671,
%U A291139 3881,3929,4289,5431,6131,6269,6491,6547,7001,7349,7583
%N A291139 Primes of the form floor(k^(3/2)) for some integer k.
%C A291139 While Piatetski-Shapiro proved that there are infinitely many primes of the form floor(n^e) with 1 < e < 12/11, it is not currently known if this sequence is infinite.
%H A291139 Charles R Greathouse IV, <a href="/A291139/b291139.txt">Table of n, a(n) for n = 1..10000</a>
%F A291139 Conjecturally, a(n) ~ (1.5 n log n)^1.5 and there are ~ x^(2/3)/log x members of this sequence up to x. - _Charles R Greathouse IV_, Oct 14 2017
%e A291139 a(1) = floor(2^(3/2)) = floor(2.8...) = 2.
%e A291139 a(2) = floor(3^(3/2)) = floor(5.1...) = 5.
%e A291139 floor(4^(3/2)) = floor(8) = 8 is composite.
%e A291139 a(3) = floor(5^(3/2)) = floor(11.1) = 11.
%e A291139 floor(6^(3/2)) = floor(14.6...) = 14 is composite.
%e A291139 floor(7^(3/2)) = floor(18.5...) = 18 is composite.
%e A291139 floor(8^(3/2)) = floor(22.6...) = 22 is composite.
%e A291139 floor(9^(3/2)) = floor(27) = 27 is composite.
%e A291139 a(4) = floor(10^(3/2)) = floor(31.6) = 31.
%t A291139 Select[Floor[Range[1000]^(3/2)],PrimeQ] (* _Harvey P. Dale_, Jul 01 2019 *)
%o A291139 (PARI) list(lim)=my(v=List(),t); lim\=1; for(n=2,sqrtnint((lim+1)^2,3)-ispower(lim+1,3), if(isprime(t=sqrtint(n^3)), listput(v,t))); Vec(v)
%Y A291139 Cf. A200141.
%K A291139 nonn
%O A291139 1,1
%A A291139 _Charles R Greathouse IV_, Aug 18 2017