A078508 Number of primes between sqrt(n^3) and sqrt((n+1)^3).

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, 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, 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

Offset

0,16

Comments

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.

Links

Table of n, a(n) for n=0..104.

Example

n = 2 [n^3/2] = 2 [(n+1)^3/2] = 5 there is 1 prime between 2 and 5 = 3.

Mathematica

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 *)

Prog

(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" "); ) ) }

Crossrefs

Sequence in context: A355343 A287160 A029443 * A029416 A252374 A344569

Adjacent sequences: A078505 A078506 A078507 * A078509 A078510 A078511

Keyword

easy,nonn

Author

Cino Hilliard, Jan 05 2003

Extensions

Comments edited by Harvey P. Dale, Jan 30 2014

Status

approved