A055071 -id:A055071 - cp's OEIS frontend

A055204 Squarefree part of n!: n! divided by its largest square divisor.

1, 2, 6, 6, 30, 5, 35, 70, 70, 7, 77, 231, 3003, 858, 1430, 1430, 24310, 12155, 230945, 46189, 969969, 176358, 4056234, 676039, 676039, 104006, 312018, 44574, 1292646, 1077205, 33393355, 66786710, 2203961430, 64822395, 90751353, 90751353

Offset: 1

Labos Elemer, Jun 19 2000

nonn

Smallest number such that n!*a(n) is a square.

			10! = 518400*7 = 7*(720)^2, so a(10) = 7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..3386 (next term has 1001 digits)
Kevin A. Broughan, Asymptotic Order of the Square-free Part of N!, Integers, 2 (2002), Article A.10.
Rafael Jakimczuk, On the h-th free part of the factorial, International Mathematical Forum, Vol. 12, No. 13 (2017), pp. 629-634.

Cf. A000142, A007913, A055071, A249831.

f[p_, e_] := p^Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40] (* Amiram Eldar, Sep 01 2024 *)
a[n_] := Block[{fi = Transpose@ FactorInteger[n!]}, Times @@ (fi[[1]]^Mod[fi[[2]], 2])]; Array[a, 40] (* Robert G. Wilson v, Nov 17 2024 *)

a(n)=core(n!) \\ Charles R Greathouse IV, Apr 03 2012

a(n) = A007913(n!) = n!/A055071(n) = A000142(n)/A055071(n).
log a(n) ~ n log 2. - Charles R Greathouse IV, Apr 03 2012
sqrt(n!) = A055772(n) * sqrt(a(n)). - Alonso del Arte, Feb 16 2015

A055773 a(n) = Product_{p in P_n} where P_n = {p prime, n/2 < p <= n }.

Original entry on oeis.org

1, 1, 2, 6, 3, 15, 5, 35, 35, 35, 7, 77, 77, 1001, 143, 143, 143, 2431, 2431, 46189, 46189, 46189, 4199, 96577, 96577, 96577, 7429, 7429, 7429, 215441, 215441, 6678671, 6678671, 6678671, 392863, 392863, 392863, 14535931, 765049, 765049, 765049

Offset: 0

Labos Elemer, Jul 12 2000

nonn

Old name: Product of primes p for which p divides n! but p^2 does not (i.e. ord_p(n!)=1). - Dion Gijswijt (gijswijt(AT)science.uva.nl), Jan 07 2007
Squarefree part of n! divided by gcd(Q,F), where Q is the largest square divisor and F is the squarefree part of n!. - Labos Elemer, Jul 12 2000
a(1) = 1, a(n) = n*a(n-1) if n is a prime else a(n) = least integer multiple of a(n-1)/n. - Amarnath Murthy, Apr 29 2004
Let P(i) denote the primorial number A034386(i). Then a(n) = P(n)/P(floor(n/2)). - Peter Luschny, Mar 05 2011
Letting H(n) = 1 + 1/2 + ... + 1/n denote the n-th harmonic number, it is known that a(n) is equal to the denominator (in lowest terms) of H(n)^2*n! for n >= 6 (see below example). - John M. Campbell, Mar 27 2016
For all n satisfying 6 <= n < 897, a(n) = A130087(n). - John M. Campbell, Mar 27 2016
It is also known that a(n) is equal to lcm^2(1, 2, ..., n)/gcd(lcm^2(1, 2, ..., n), n!). - John M. Campbell, Apr 04 2016

			n = 13, P_n = {7, 11, 13}, a(13) = 7*11*13 = 1001.
Letting n = 14, the denominator (in lowest terms) of H(n)^2*n! = 131803989435744/143 is a(14)=143. - _John M. Campbell_, Mar 27 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200 (corrected by Michel Marcus, Jan 19 2019)
J. M. Campbell et al., A problem involving the product prod_{k=1..n} k^mu(k), where mu denotes the Möbius function, Mathematics Stack Exchange (2016).
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
Index entries for sequences related to Gijswijt's sequence

Cf. A000188, A008833, A007913, A055229, A055231 (for n), A055071, A055204, A055230, A094299, A094302, A193477, A130087.

a := n -> mul(k,k=select(isprime,[$iquo(n,2)+1..n])); # Peter Luschny, Jun 20 2009
A055773 := n -> numer(n!/iquo(n,2)!^4); # Peter Luschny, Jul 30 2011

Table[Numerator[n!/Floor[n/2]!^4], {n, 0, 40}] (* Michael De Vlieger, Mar 27 2016 *)

q=1;for(n=2,41,print1(q,",");q=if(isprime(n),q*n,q/gcd(q,n))) \\ Klaus Brockhaus, May 02 2004

a(n) = k=1;forprime(p=nextprime(n\2+1),precprime(n),k=k*p);k \\ Klaus Brockhaus, May 02 2004

a(n) = prod(i=primepi(n/2)+1,primepi(n),prime(i)) \\ John M. Campbell, Mar 27 2016

from math import prod
from sympy import primerange
def A055773(n): return prod(primerange((n>>1)+1,n+1)) # Chai Wah Wu, Apr 13 2024

a(n) = numerator(A056040(n)^2/n!).
a(n) = numerator(A056040(n)/floor(n/2)!^2).
a(n) = numerator(n!/floor(n/2)!^4). - Peter Luschny, Jul 30 2011
a(n) = product of primes p such that n/2 < p <= n. - Klaus Brockhaus, May 02 2004
a(n) = A055204(n)/A055230(n) = A055231(n!) = A007913(n!)/A055229(n!).
a(n) = Product_{i=pi(n/2)+1..pi(n)} prime(i), where pi denotes the prime counting function and prime(i) denotes the i-th prime number. - John M. Campbell, Mar 27 2016

Entry revised by N. J. A. Sloane, Jan 07 2007

Simpler definition based on a comment of Klaus Brockhaus, set offset to 0 and prepended 1 to data. - Peter Luschny, Mar 09 2013

A055772 Square root of largest square dividing n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 12, 12, 24, 72, 720, 720, 1440, 1440, 10080, 30240, 120960, 120960, 725760, 725760, 7257600, 7257600, 79833600, 79833600, 958003200, 4790016000, 62270208000, 186810624000, 2615348736000, 2615348736000, 15692092416000

Offset: 1

Labos Elemer, Jul 12 2000

nonn

			For n=6, 6! = 720 = 144*5 so a(6) = sqrt(144) = 12.

Charles R Greathouse IV, Table of n, a(n) for n = 1..500

Cf. A000188, A008833, A007913, A055229, A055231, A055071, A055204, A055230.

a:= proc(n)
local r,F,t;
r:= n!;
F:= ifactors(r)[2];
mul(t[1]^floor(t[2]/2),t=F)
end proc:
seq(a(n), n= 1 .. 100); # Robert Israel, Oct 19 2014

Table[Last[Select[Sqrt[#]&/@Divisors[n!],IntegerQ]],{n,30}] (* Harvey P. Dale, Oct 08 2012 *)
(Sqrt@Factorial@Range@30)/.Sqrt[]->1 (* _Morgan L. Owens, May 04 2016 *)
f[p_, e_] := p^Floor[e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40] (* Amiram Eldar, Jul 26 2024 *)

a(n)=core(n!,2)[2] \\ Charles R Greathouse IV, Apr 03 2012

a(n) = A000188(n!) = sqrt(A008833(n!)) = sqrt(A055071(n)).
n! = a(n)^2*A055204(n) = a(n)^2*A007913(n!).
n! = (A000188(n!)^2)*A055229(n!)*A055231(n!).
log(a(n)) ~ n*log(n)/2. - David Radcliffe, Oct 17 2014

A353897 a(n) is the largest divisor of n whose exponents in its prime factorization are all powers of 2 (A138302).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 16, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 16, 65, 66, 67, 68

Offset: 1

Amiram Eldar, May 10 2022

			a(27) = 9 since 9 = 3^2 is the largest divisor of 27 with an exponent in its prime factorization, 2, that is a power of 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A138302, A353898.

Similar sequences: A000265, A007947, A008834, A055071, A350390.

f[p_, e_] := p^(2^Floor[Log2[e]]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Multiplicative with a(p^e) = p^(2^floor(log_2(e))).
a(n) = n if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c*n^2, where c = 0.4616988732... = (1/2) * Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^(f(k-1)+1))/p^(2*k)), f(k) = 2^floor(log_2(k)) and f(0) = 0.

A055230 Greatest common divisor of largest square dividing n! and squarefree part of n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 6, 10, 10, 10, 5, 5, 1, 21, 42, 42, 7, 7, 14, 42, 6, 6, 5, 5, 10, 330, 165, 231, 231, 231, 462, 2002, 5005, 5005, 4290, 4290, 390, 78, 39, 39, 13, 13, 26, 1326, 102, 102, 17, 935, 13090, 746130, 373065, 373065, 24871, 24871

Offset: 1

Labos Elemer, Jun 21 2000

nonn

			a(5) = 2 because 5! = 120; largest square divisor is 4, squarefree part is 30; GCD(4, 30) = 2.
a(7) = 1 because 7! = 5040; the largest square divisor is 144 and the squarefree part is 35 and these are coprime.

Michael De Vlieger, Table of n, a(n) for n = 1..5000

Cf. A008833, A007913, A000188, A000720.

Table[GCD[Times @@ Flatten@ Map[Table[#1, 2 Floor[#2/2]] & @@ # &, #], Times @@ Flatten@ Map[Table[#1, Floor[Mod[#2, 2]]] & @@ # &, #]] &@ FactorInteger[n!], {n, 61}] (* Michael De Vlieger, Jul 26 2016 *)

a(n) = my(fn=n!, cn=core(fn)); gcd(cn, fn/cn); \\ Michel Marcus, Dec 10 2013

a(n) = GCD(A008833(n!), A007913(n!)) = GCD(A055071(n), A055204(n)).

A065886 Smallest square divisible by n!.

Original entry on oeis.org

1, 1, 4, 36, 144, 3600, 3600, 176400, 2822400, 25401600, 25401600, 3073593600, 110649369600, 18699743462400, 74798973849600, 1869974346240000, 29919589539840000, 8646761377013760000, 77820852393123840000, 28093327713917706240000, 112373310855670824960000

Offset: 0

Henry Bottomley, Nov 27 2001

nonn

			a(10) = 25401600 since 10! = 3628800 and the smallest square divisible by this is 25401600 = 3628800*7 = 5040^2

Robert Israel, Table of n, a(n) for n = 0..407

N:= 50: # to get a(0)..a(N)
P:= select(isprime, [$2..N]):
nP:= nops(P):
V:= Vector(nP):
A[0]:= 1:
for n from 1 to N do
  for i from 1 to nP do V[i]:= V[i] + padic:-ordp(n,P[i]) od;
  A[n]:= mul(P[i]^(2*ceil(V[i]/2)),i=1..nP)
od:
seq(A[n],n=0..N); # Robert Israel, Jan 30 2017

ssd[n_]:=Module[{nf=n!,k=1},While[!IntegerQ[Sqrt[k*nf]],k++];k*nf]; Array[ssd,20,0] (* Harvey P. Dale, Apr 29 2012 *)

a(n) = A053143(A000142(n)) = A065887(n)^2 = A000142(n)*A055204(n) = A001044(n)/A055071(n)

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.

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

cp's OEIS Frontend

A055204 Squarefree part of n!: n! divided by its largest square divisor.

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A055773 a(n) = Product_{p in P_n} where P_n = {p prime, n/2 < p <= n }.

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A055772 Square root of largest square dividing n!.

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A353897 a(n) is the largest divisor of n whose exponents in its prime factorization are all powers of 2 (A138302).

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A055230 Greatest common divisor of largest square dividing n! and squarefree part of n!.

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A065886 Smallest square divisible by n!.

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A374988 Largest unitary square divisor of n!.

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