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

0,7

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

Formula

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

a(n) = A071974(n!).

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]

Prog

(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

Crossrefs

Cf. A006364, A055772, A071974, A374988.

Sequence in context: A125509 A281251 A247511 * A097824 A133208 A118877

Adjacent sequences: A374986 A374987 A374988 * A374990 A374991 A374992

Keyword

nonn

Author

Amiram Eldar, Jul 26 2024

Status

approved