A374989 Square root of largest unitary square divisor of n!.

1, 1, 1, 1, 1, 1, 12, 12, 3, 9, 720, 720, 160, 160, 35, 189, 189, 189, 145152, 145152, 7257600, 12800, 275, 275, 136857600, 684288000, 4343625, 17875, 875875, 875875, 125536739328, 125536739328, 15324309, 637, 709689344, 9052160000, 18104320000, 18104320000, 2624375

Offset: 0

Author

Amiram Eldar, Jul 26 2024

Keywords

• nonn

Comments

Unitary analog of A055772.

a(n) is even if and only if n > 1 and is in A006364.

Links

• Amiram Eldar, Table of n, a(n) for n = 0..1118

Crossrefs

Cf. A006364, A055772, A071974, A374988.

Programs

Mathematica

f[p_, e_] := If[EvenQ[e], p^(e/2), 1]; a[0] = a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40, 0]

PARI

a(n) = {my(f = factor(n!)); prod(i = 1, #f~, if(f[i, 2]%2, 1, f[i, 1]^(f[i, 2]/2)));}

Python

from math import prod
from itertools import count, islice
from collections import Counter
from sympy import factorint
def A374989_gen(): # generator of terms
    c = Counter()
    for i in count(0):
        c += Counter(factorint(i))
        yield prod(p**(e>>1) for p, e in c.items() if e&1^1)
A374989_list = list(islice(A374989_gen(),30)) # Chai Wah Wu, Jul 27 2024

Formula

a(n) = sqrt(A374988(n)).

a(n) = A071974(n!).