A381665 Integers k such that prime(k)!/k^k is an integer.

1, 12, 24, 36, 40, 45, 48, 60, 72, 80, 90, 96, 120, 144, 160, 180, 192, 210, 216, 224, 240, 252, 270, 280, 288, 315, 320, 336, 360, 378, 420, 432, 448, 480, 504, 540, 560, 576, 630, 640, 672, 720, 756, 840, 864, 896, 945, 960, 1008, 1080, 1120, 1134, 1152, 1200, 1260, 1280, 1296

Offset: 1

Author

Michel Marcus, Mar 03 2025

Keywords

• nonn

Links

• Michel Marcus, Table of n, a(n) for n = 1..211

Crossrefs

Cf. A000312, A039716.

Programs

Mathematica

Select[Range[1296],IntegerQ[Prime[#]!/#^#]&] (* James C. McMahon, Mar 03 2025 *)

PARI

isok(k) = Mod(prime(k)!, k^k) == 0;

Python

from collections import Counter
from itertools import count, islice
from sympy import prime, factorint
def A381665_gen(): # generator of terms
    c, p = Counter(), 1
    for k in count(1):
        q, m = prime(k), Counter({a:b*k for a, b in factorint(k).items()})
        c += sum((Counter(factorint(i)) for i in range(p+1,q+1)),start=Counter())
        if m<=c:
            yield k
        p = q
A381665_list = list(islice(A381665_gen(),57)) # Chai Wah Wu, Mar 03 2025