A055772 -id:A055772 - cp's OEIS frontend

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

Amiram Eldar, Jul 26 2024

nonn

Unitary analog of A055772.

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

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

Cf. A006364, A055772, A071974, A374988.

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]

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

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

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

a(n) = A071974(n!).

A056043 Let k be largest number such that k^2 divides n!; a(n) = k/floor(n/2)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 6, 6, 2, 2, 2, 6, 3, 3, 2, 2, 2, 2, 2, 2, 2, 10, 10, 30, 30, 30, 12, 12, 3, 3, 6, 30, 10, 10, 10, 30, 6, 6, 2, 2, 2, 30, 60, 60, 30, 210, 42, 42, 42, 42, 28, 28, 2, 2, 4, 4, 4, 4, 4, 84, 21, 21, 14, 14, 14, 42, 6, 6, 2, 2, 2, 10, 10, 70, 140, 140, 14, 126, 126

Offset: 1

Labos Elemer, Jul 25 2000

nonn

			For n = 7, 7! = 5040 = 144*35, so 12 is its largest square-root-divisor, A000188(5040), and it is divisible by 6 = 3!, so a(7) = 12/3! = 2.

Cf. A000142, A000188, A001405, A004526, A055772, A081123.

f[p_, e_] := p^Floor[e/2]; b[1] = 1; a[n_] := (Times @@ f @@@ FactorInteger[n!]) / Floor[n/2]!; Array[a, 100] (* Amiram Eldar, May 24 2024 *)

a(n) = A000188(n!)/floor(n/2)! = A055772(n)/A000142(A004526(n)) = A055772(n)/A081123(n). [Corrected by Amiram Eldar, May 24 2024]

A056194 Characteristic cube divisor of n!: a(n) = A056191(n!).

Original entry on oeis.org

1, 1, 1, 8, 8, 1, 1, 8, 8, 1, 1, 27, 27, 216, 1000, 1000, 1000, 125, 125, 1, 9261, 74088, 74088, 343, 343, 2744, 74088, 216, 216, 125, 125, 1000, 35937000, 4492125, 12326391, 12326391, 12326391, 98611128, 8024024008, 125375375125

Offset: 1

Labos Elemer, Aug 02 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..4054

Cf. A000142, A055229, A000188, A008833, A007913, A055231, A055230, A055772, A055071, A055204, A055773 A056552, A056195.

f[p_, e_] := If[OddQ[e] && e > 1, p^3, 1]; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40] (* Amiram Eldar, Sep 06 2020 *)

a(n) = A056191(A000142(n)). - Amiram Eldar, Sep 06 2020

A056195 a(n) = n! divided by its characteristic cube divisor A056194.

Original entry on oeis.org

1, 2, 6, 3, 15, 720, 5040, 5040, 45360, 3628800, 39916800, 17740800, 230630400, 403603200, 1307674368, 20922789888, 355687428096, 51218989645824, 973160803270656, 2432902008176640000, 5516784599040000

Offset: 1

Labos Elemer, Aug 02 2000

nonn

			n = 10, a(10) = 10! because g(10!) = 1.
n = 9, a(9) = 45320 because 9! = 2*2*2*2*2*2*2*3*3*5*7 and g(9!) = 2, so a(9) = 9!/8.

Amiram Eldar, Table of n, a(n) for n = 1..491

Cf. A055229, A055230, A055071, A055772, A055773.

f[p_, e_] := If[OddQ[e] && e > 1, p^3, 1]; a[n_] := n! / (Times @@ f @@@ FactorInteger[n!]); Array[a, 20] (* Amiram Eldar, Sep 06 2020 *)

The CCD of n! is the cube of g = A055229(n!) = A055230(n). So a(n) = n!/ggg = L*L*f where L = A000188(n!)/A055229(n!) = A055772(n)/A055230(n) and f = A055231(n!) = A055773(n).

A105350 Largest squared factorial dividing n!.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 36, 36, 576, 576, 518400, 518400, 518400, 518400, 25401600, 25401600, 1625702400, 1625702400, 131681894400, 131681894400, 13168189440000, 13168189440000, 1593350922240000, 1593350922240000, 229442532802560000, 229442532802560000

Offset: 0

Reinhard Zumkeller, Apr 01 2005

nonn

a(n) = A001044(A056039(n)) = A056038(n)^2.

Whenever n > 1 is not in A056067, a(n) = A180064(n). - Andrey Zabolotskiy, Oct 19 2023

Index entries for sequences related to factorial numbers.

Cf. A000142, A000290, A055071, A055772, A056067.

Cf. A180064.

a[n_] := (For[k = 1, Divisible[n!, k!^2], k++]; (k-1)!^2)
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Aug 07 2018 *)

Data and offset corrected by Jean-François Alcover, Aug 07 2018

Edited by Andrey Zabolotskiy, Oct 18 2023

A055928 Sum of square divisors of n! = sum of squares of divisors of the square root of largest square dividing n!.

Original entry on oeis.org

1, 1, 1, 5, 5, 210, 210, 850, 7735, 806806, 806806, 3229590, 3229590, 161479500, 1455090000, 23286770000, 23286770000, 838446909300, 838446909300, 83973923013750, 83973923013750, 10244818607677500, 10244818607677500

Offset: 1

Labos Elemer, Jul 21 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..505

A001157, A000142, A035316, A055772, A000188.

f[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p^2-1) , (p^(e+2)-1)/(p^2-1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n!]); Array[a, 23] (* Amiram Eldar, Aug 01 2019 *)

a(n) = A035316(n!) = A001157(A055772(n)) = A001157(A000188(n!)).

A056596 Number of nonsquare divisors of n!.

Original entry on oeis.org

0, 1, 3, 6, 14, 24, 54, 88, 148, 240, 510, 756, 1548, 2520, 3936, 5248, 10624, 14508, 29196, 40740, 60500, 95400, 191400, 242016, 338880, 529920, 674688, 912912, 1830192, 2327424, 4660224, 5523456, 7858176, 12152064, 16406592, 19576080

Offset: 1

Labos Elemer, Jul 21 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

A000142, A000005, A046951, A055772, A027423, A055993.

[#[d:d in Divisors(Factorial(n))| not IsSquare(d)]:n in [1..36]]; // Marius A. Burtea, Jul 16 2019

Table[Count[Divisors[n!], d_ /; !IntegerQ@ Sqrt@ d], {n, 30}] (* Amiram Eldar, Jul 16 2019 after Michael De Vlieger at A055993 *)

a(n) = d(n!) - A046951(n!)

a(n) = A027423(n) - A055993(n). - Amiram Eldar, Jul 16 2019

A056628 a(n) = A056623(n!).

Original entry on oeis.org

1, 1, 1, 1, 1, 144, 144, 144, 1296, 518400, 518400, 230400, 230400, 2822400, 9144576, 146313216, 146313216, 21069103104, 21069103104, 52672757760000, 119439360000, 3613040640000, 3613040640000, 18730002677760000, 468250066944000000, 19783565328384000000, 19783565328384000000

Offset: 1

Labos Elemer, Aug 08 2000

nonn

Previous name "Largest unitary square divisor of n!" was incorrect. See A374988 for the correct sequence with this name. - Amiram Eldar, Jul 26 2024

			a(12) = A056623(12!) = A008833(12!)/A055229(12!)^2 = 2073600/3^2 = 230400.

Cf. A055772, A055071, A055230, A000188, A008833, A055229, A056623, A056627, A374988.

f[p_, 1] := 1; f[p_, e_] := If[EvenQ[e], p^e, p^(e-3)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 27] (* 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) = A056623(n!); \\ Michel Marcus, Aug 16 2020

a(n) = A055071(n)/A055230(n)^2 = A008833(n!)/A055229(n!)^2.
a(n) = A056623(n!). - Michel Marcus, Aug 16 2020
a(n) = A056627(n)^2. - 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

cp's OEIS Frontend

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

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A056043 Let k be largest number such that k^2 divides n!; a(n) = k/floor(n/2)!.

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A056194 Characteristic cube divisor of n!: a(n) = A056191(n!).

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A056195 a(n) = n! divided by its characteristic cube divisor A056194.

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A105350 Largest squared factorial dividing n!.

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A055928 Sum of square divisors of n! = sum of squares of divisors of the square root of largest square dividing n!.

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A056596 Number of nonsquare divisors of n!.

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A056628 a(n) = A056623(n!).

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