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A374989
Square root of largest unitary square divisor of n!.
3
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 26 2024
STATUS
approved