A229836 Number of primes between n! and n^n inclusive.

0, 2, 6, 45, 415, 4693, 65010, 1073640, 20669837, 454793822, 11259684418, 309761863916, 9373389023182, 309374515194621, 11059527891811334, 425655578031419604, 17547665070746310736, 771403345825446116583, 36020103485009885093324

Offset

1,2

Links

Table of n, a(n) for n=1..19.

Formula

a(n) = A064151(n) - A003604(n). Add 1 for n = 2 since 2! is prime. - Jens Kruse Andersen, Jul 29 2014

Example

There are 45 primes between 4! = 24 and 4^4 = 256.

Maple

with(numtheory): A229836:=n->pi(n^n)-pi(n!): (0, 2, seq(A229836(n), n=3..10)); # Wesley Ivan Hurt, Nov 17 2015

Mathematica

Join[{0, 2}, Table[PrimePi[n^n] - PrimePi[n!], {n, 3, 12}]] (* Wesley Ivan Hurt, Nov 17 2015 *)

Prog

(Python)
import math
import sympy
from sympy import sieve
x = 1
while x < 50:
....y = [i for i in sieve.primerange(math.factorial(x), x**x)]
....print(len(y))
....x += 1
(Python)
from math import factorial
from sympy import primepi
def A229836(n): return primepi(n**n)-primepi(factorial(n)-1) # Chai Wah Wu, Jun 06 2024
(PARI) a(n)=primepi(n^n)-primepi(n!-1) \\ Charles R Greathouse IV, Apr 30 2014
(PARI) a(n) = if(n==2, 2, primepi(n^n)-primepi(n!)) \\ Altug Alkan, Nov 17 2015

Crossrefs

Cf. A000142, A000312, A003604, A064151.

Sequence in context: A332757 A255015 A347984 * A359659 A374874 A370818

Adjacent sequences: A229833 A229834 A229835 * A229837 A229838 A229839

Keyword

nonn,more

Author

Derek Orr, Dec 30 2013

Extensions

a(12)-a(16) from Jens Kruse Andersen, Jul 29 2014

a(17)-a(18) from Chai Wah Wu, Jun 06 2024

a(19) from Amiram Eldar, Jun 11 2024

Status

approved