A053810 Numbers of the form p^e where both p and e are prime numbers.

4, 8, 9, 25, 27, 32, 49, 121, 125, 128, 169, 243, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2048, 2187, 2197, 2209, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 8192, 9409, 10201, 10609, 11449, 11881, 12167

Offset: 1

Author

Henry Bottomley, Mar 28 2000

Keywords

• easy
• nonn

Comments

Possible orders of finite fields with exactly 2 subfields. In other words, possible orders of finite fields whose only proper subfield is the prime field. - Jianing Song, Jun 06 2025

Links

• T. D. Noe, Table of n, a(n) for n = 1..9965

Crossrefs

Cf. A000040, A000961, A053811, A053812.

Cf. A203967; subsequence of A000961.

Cf. A113877 (similar for semiprimes).

Programs

Haskell

a053810 n = a053810_list !! (n-1)
a053810_list = filter ((== 1) . a010051 . a100995) $ tail a000961_list
-- Reinhard Zumkeller, Jun 05 2013

Maple

h := proc(n) local P; P := NumberTheory:-PrimeFactors(n); nops(P) = 1 and isprime(padic:-ordp(n, P[1])) end:
A053810List := upto -> seq(n, n = select(h, [seq(1..upto)])):  # Peter Luschny, Apr 14 2025

Mathematica

pp={}; Do[if=FactorInteger[n]; If[Length[if]==1&&PrimeQ[if[[1, 1]]]&&PrimeQ[if[[1, 2]]], pp=Append[pp, n]], {n, 13000}]; pp
Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[12800]]}, {i, 1, PrimePi[ Log[ Prime[n], 12800]]}]]]

PARI

is(n)=isprime(isprimepower(n)) \\ Charles R Greathouse IV, Mar 19 2013

Python

from sympy import primepi, integer_nthroot, primerange
def A053810(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x, p)[0]) for p in primerange(x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 13 2024

SageMath

def isA(n):
    p = prime_divisors(n)
    return len(p) == 1 and is_prime(valuation(n, p[0]))
print([n for n in srange(1, 12222) if isA(n)])  # Peter Luschny, Apr 14 2025

Formula

a(n) = A053811(n)^A053812(n). - David Wasserman, Feb 17 2006

A010055(a(n)) * A010051(A100995(a(n))) = 1. - Reinhard Zumkeller, Jun 05 2013

Sum_{n>=1} 1/a(n) = Sum_{p prime} P(p) = 0.6716752222..., where P is the prime zeta function. - Amiram Eldar, Nov 21 2020

Extensions

More terms from David Wasserman, Feb 17 2006

Name clarified by Peter Luschny, Apr 14 2025