A374989 -id:A374989 - cp's OEIS frontend

A056627 a(n) = A056622(n!).

1, 1, 1, 1, 1, 12, 12, 12, 36, 720, 720, 480, 480, 1680, 3024, 12096, 12096, 145152, 145152, 7257600, 345600, 1900800, 1900800, 136857600, 684288000, 4447872000, 4447872000, 435891456000, 435891456000, 3138418483200, 3138418483200, 6276836966400, 190207180800

Offset: 1

Labos Elemer, Aug 08 2000

nonn

Previous name "Square root of largest unitary square divisor of n!" was incorrect. See A374989 for the correct sequence with this name. - Amiram Eldar, Jul 26 2024

			a(12) = A056622(12!) = A000188(12!)/A055229(12!) = 1440/3 = 480.

Cf. A055772, A055230, A000188, A055229, A056622, A056628, A374989.

f[p_, 1] := 1; f[p_, e_] := If[EvenQ[e], p^(e/2), p^((e-3)/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 33] (* Amiram Eldar, Jul 08 2024 *)

A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ Charles R Greathouse IV, Nov 20 2012
A008833(n) = n/core(n) \\ Michael B. Porter, Oct 17 2009
A056623(n) = (A008833(n)/(A055229(n)^2)); \\ Antti Karttunen, Nov 19 2017
a(n) = sqrtint(A056623(n!)); \\ Michel Marcus, Aug 16 2020

a(n) = A055772(n)/A055230(n) = A000188(n!)/A055229(n!).
a(n) = A056622(n!). - Michel Marcus, Aug 16 2020
a(n) = sqrt(A056628(n)). - Amiram Eldar, Jul 08 2024

More terms from Michel Marcus, Aug 16 2020

Incorrect name replaced with a formula by Amiram Eldar, Jul 26 2024

A374988 Largest unitary square divisor of n!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 144, 144, 9, 81, 518400, 518400, 25600, 25600, 1225, 35721, 35721, 35721, 21069103104, 21069103104, 52672757760000, 163840000, 75625, 75625, 18730002677760000, 468250066944000000, 18867078140625, 319515625, 767157015625, 767157015625, 15759472921106221891584

Offset: 0

Amiram Eldar, Jul 26 2024

nonn

Unitary analog of A055071.

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

Amiram Eldar, Table of n, a(n) for n = 0..673

Cf. A006364, A055071, A350388, A374989.

f[p_, e_] := If[EvenQ[e], p^e, 1]; a[0] = a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30, 0]

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

from math import prod
from itertools import count, islice
from collections import Counter
from sympy import factorint
def A374988_gen(): # generator of terms
    c = Counter()
    for i in count(0):
        c += Counter(factorint(i))
        yield prod(p**e for p, e in c.items() if e&1^1)
A374988_list = list(islice(A374988_gen(),30)) # Chai Wah Wu, Jul 27 2024

a(n) = A350388(n!).

a(n) = A374989(n)^2.

cp's OEIS Frontend

A056627 a(n) = A056622(n!).

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