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

2, 5, 11, 31, 41, 89, 103, 181, 281, 311, 353, 419, 769, 797, 811, 839, 853, 911, 1091, 1153, 1201, 1217, 1249, 1499, 1621, 1873, 2081, 2557, 2999, 3307, 3533, 3671, 3881, 3929, 4289, 5431, 6131, 6269, 6491, 6547, 7001, 7349, 7583

Offset

1,1

Comments

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.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

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

Example

a(1) = floor(2^(3/2)) = floor(2.8...) = 2.
a(2) = floor(3^(3/2)) = floor(5.1...) = 5.
floor(4^(3/2)) = floor(8) = 8 is composite.
a(3) = floor(5^(3/2)) = floor(11.1) = 11.
floor(6^(3/2)) = floor(14.6...) = 14 is composite.
floor(7^(3/2)) = floor(18.5...) = 18 is composite.
floor(8^(3/2)) = floor(22.6...) = 22 is composite.
floor(9^(3/2)) = floor(27) = 27 is composite.
a(4) = floor(10^(3/2)) = floor(31.6) = 31.

Mathematica

Select[Floor[Range[1000]^(3/2)], PrimeQ] (* Harvey P. Dale, Jul 01 2019 *)

Prog

(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)

Crossrefs

Cf. A200141.

Sequence in context: A292210 A079225 A139466 * A139467 A124481 A002862

Adjacent sequences: A291136 A291137 A291138 * A291140 A291141 A291142

Keyword

nonn

Author

Charles R Greathouse IV, Aug 18 2017

Status

approved