A051696 -id:A051696 - cp's OEIS frontend

A067911 Product of gcd(k,n) for 1 <= k <= n.

1, 2, 3, 8, 5, 72, 7, 128, 81, 800, 11, 41472, 13, 6272, 30375, 32768, 17, 3359232, 19, 20480000, 750141, 247808, 23, 13759414272, 15625, 1384448, 1594323, 5035261952, 29, 30233088000000, 31, 2147483648, 235782657, 37879808

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Mar 10 2002

nonn

T. D. Noe, Table of n, a(n) for n = 1..500
Johann Cigler, Some remarks on the power product expansion of the q-exponential series, arXiv:2006.06242 [math.CO], 2020.
Gottfried Helms, A dream of a (number-) sequence, 2007-2009.
L. Toth, A survey of gcd-sum functions, J. Int. Seq. 13 (2010) # 10.8.1.

Cf. A051190, A051696.
In A018804 the product is replaced by sum.
Product of terms in n-th row of A050873.
Cf. A000010 (comments on product formulas).

with(numtheory): a := n -> mul(d^phi(n/d), d = divisors(n)):
seq(a(i), i = 1..34); # Peter Luschny, Apr 07 2013

a[n_] := Product[d^EulerPhi[n/d], {d, Divisors[n]}];
Array[a, 34] (* Jean-François Alcover, Jun 03 2019 *)

a(n) = prod(k=1, n, gcd(k, n)); \\ Michel Marcus, Aug 23 2016

A067911 = lambda n: mul(gcd(n,i) for i in range(n))
[A067911(n) for n in (1..34)] # Peter Luschny, Apr 07 2013

a(n) = Product_{d|n} d^phi(n/d). - Vladeta Jovovic, Mar 08 2004
a(n) = n*A051190(n). - Peter Luschny, Apr 07 2013
a(n) = Product_{k=1..n} (n/gcd(n,k))^(phi(gcd(n,k))/phi(n/gcd(n,k))) where phi = A000010. - Richard L. Ollerton, Nov 07 2021

Extended and edited by John W. Layman, Mar 14 2002

A095996 a(n) = largest divisor of n! that is coprime to n.

Original entry on oeis.org

1, 1, 2, 3, 24, 5, 720, 315, 4480, 567, 3628800, 1925, 479001600, 868725, 14350336, 638512875, 20922789888000, 14889875, 6402373705728000, 14849255421, 7567605760000, 17717861581875, 1124000727777607680000, 2505147019375

Offset: 1

Robert G. Wilson v, Jul 19 2004, based on a suggestion from Leroy Quet, Jun 18 2004

nonn

The denominators of the coefficients in Taylor series for LambertW(x) are 1, 1, 1, 2, 3, 24, 5, 720, 315, 4480, 567, 3628800, 1925, ..., which is this sequence prefixed by 1. (Cf. A227831.) - N. J. A. Sloane, Aug 02 2013

The second Mathematica program is faster than the first for large n. - T. D. Noe, Sep 07 2013

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., Eq. (5.66).

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A036503, A227831, A066570.

[Denominator(n^n/Factorial(n)): n in [1..25]]; // Vincenzo Librandi, Sep 04 2014

series(LambertW(x),x,30); # N. J. A. Sloane, Jan 08 2021

f[n_] := Select[Divisors[n! ], GCD[ #, n] == 1 &][[ -1]]; Table[f[n], {n, 30}]
Denominator[Exp[Table[Limit[Zeta[s]*Sum[(1 - If[Mod[k, n] == 0, n, 0])/k^(s - 1), {k, 1, n}], s -> 1], {n, 1, 30}]]] (* Conjecture Mats Granvik, Sep 09 2013 *)
Table[Denominator[n^n/n!], {n, 30}] (* Vincenzo Librandi, Sep 04 2014 *)

a(n):=sum((-1)^(n-j)*binomial(n,j)*(j/n+1)^n,j,0,n);
makelist(num(a(n)),n,1,20); /* Vladimir Kruchinin, Jun 02 2013 */

a(n) = denominator(n^n/n!); \\ G. C. Greubel, Nov 14 2017

a(p) = (p-1)!.
a(n) = n!/A051696(n) = (n-1)!/A062763(n).
a(n) = numerator(Sum_{j = 0..n} (-1)^(n-j)*binomial(n,j)*(j/n+1)^n ). - Vladimir Kruchinin, Jun 02 2013
a(n) = denominator(n^n/n!). - Vincenzo Librandi Sep 04 2014

A064446 a(n) = gcd(n!, n^n, lcm(1, 2, ..., n)), or gcd(n^n, lcm(1, 2, ..., n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 7, 8, 9, 40, 11, 72, 13, 56, 45, 16, 17, 144, 19, 80, 63, 176, 23, 144, 25, 208, 27, 112, 29, 10800, 31, 32, 297, 544, 175, 864, 37, 608, 351, 800, 41, 6048, 43, 352, 675, 736, 47, 864, 49, 800, 459, 416, 53, 864, 275, 1568, 513, 928, 59, 21600, 61

Offset: 1

Labos Elemer, Oct 02 2001

nonn

gcd(n^n, lcm(1..n)) must be limited to products of all the distinct prime divisors p of n. We can regard lcm(1..n) as the product of a "regular" factor r produced by primes that also divide n and a coprime factor s produced by primes that are coprime to n. Since the distinct prime divisors p of n are the only distinct prime divisors of n^n, we need only consider r and can ignore s when considering gcd(n^n, lcm(1..n)). Because r is the product of the largest power e_1 of each distinct prime divisor p, and since the power e_2 of the corresponding primes that divide n^n must always be such that e_2 >= e_1, it is sufficient to compute r to determine a(n). - Michael De Vlieger, Oct 26 2015

			n=6: a(6) = gcd(720, 60, 46656) = 12.
Since only 1 and 5 are relatively prime to 6, a(6) = lcm(1,2,3,4,5,6) / lcm(1,5) = 60/5 = 12.

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (First 1000 terms from Harry J. Smith)

Cf. A000142, A000312, A051696. Equals A003418(n)/A038610(n).

List([1..70],n->Gcd(Factorial(n),n^n,Lcm([1..n]))); # Muniru A Asiru, Mar 20 2018

A064446 := n -> ilcm(seq(i,i=1..n))/ilcm(op(select(k->igcd(n,k)=1,[$1..n])));
seq(A064446(i),i=0..61); # Peter Luschny, Jun 25 2011
N:= 1000: # to get a(1) to a(N)
Primes:= select(isprime, [2,seq(2*i+1,i=1..floor((N-1)/2))]):
A:= Vector(N,1):
for p in Primes do
  for d from 1 to floor(log[p](N)) do
    for j from p^d to min(N, p^(d+1)-p) by p do
       A[j]:= A[j]*p^d
od od od:
convert(A,list); # Robert Israel, Oct 26 2015

Table[GCD[n!,n^n,LCM@@Range[n]],{n,70}] (* Harvey P. Dale, Jun 25 2011 *)
f[n_] := Block[{p = First /@ FactorInteger@ n}, Times @@ Power @@@ Transpose[{p, Floor@ Log[#, n] & /@ p}]]; {1}~Join~Table[f@ n, {n, 2, 10000}] (* Michael De Vlieger, Oct 26 2015 *)

L=1; for (n=1, 1000, L=lcm(L, n); write("b064446.txt", n, " ", gcd(n^n, L))) \\ Harry J. Smith, Sep 14 2009

a(n)=my(f=factor(n)); for(i=1,#f~, f[i,2]=logint(n,f[i,1])); factorback(f) \\ Charles R Greathouse IV, Nov 19 2015

a(n) = gcd(n^n, lcm(vector(n, k, k))); \\ Michel Marcus, Mar 18 2018

a(n) = gcd(A000142(n), A000312(n), A003418(n)) = gcd(A000312(n), A003418(n)) = gcd(A051696(n), A003418(n)).
a(n) = Product_{prime p | n} p^floor(log_p(n)). - Robert Israel, Oct 26 2015
a(n) = e^(Sum_{k=1..n} (floor(n^k/k) - floor((n^k - 1)/k))*Mangoldt(k)) where Mangoldt is the Mangoldt function. - Anthony Browne, Jun 16 2016

A062763 a(n) is the greatest common divisor of (n-1)! and n^n.

Original entry on oeis.org

1, 1, 1, 2, 1, 24, 1, 16, 9, 640, 1, 20736, 1, 7168, 6075, 2048, 1, 23887872, 1, 8192000, 321489, 2883584, 1, 10319560704, 625, 54525952, 59049, 2877292544, 1, 835884417024000000, 1, 67108864, 578739249, 36507222016, 187578125, 61628086298345472, 1

Offset: 1

Henry Bottomley, Jul 16 2001

nonn

a(n) = 1 iff n is 1 or a prime.

			a(10) = gcd(9!, 10^10) = gcd(2^7*3^4*5*7, 2^10*5^10) = 2^7*5 = 640.

Harry J. Smith, Table of n, a(n) for n = 1..300

Cf. A051696.

a(n)=gcd((n-1)!, n^n); \\ Harry J. Smith, Aug 10 2009

a(n) = A051696(n)/n.

A055774 Least common multiple of n! and n^n.

Original entry on oeis.org

1, 4, 54, 768, 75000, 233280, 592950960, 5284823040, 1735643790720, 5670000000000, 1035338990313196800, 17163493362892800, 145077660657859734604800, 9653278129532887449600

Offset: 1

Henry Bottomley, Jul 12 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000142 (n!), A000312 (n^n), A051696 (gcd).

List([1..200], n->Lcm(Factorial(n),n^n)); # Muniru A Asiru, Feb 04 2018

A055774:=n->lcm(n!,n^n): seq(A055774(n), n=1..15); # Wesley Ivan Hurt, Jul 07 2014

Table[LCM[n!, n^n], {n, 15}] (* Wesley Ivan Hurt, Jul 07 2014 *)

a(n) = lcm(n!, n^n); \\ Michel Marcus, Feb 06 2018

a(n) = lcm(A000312(n), A000142(n)) = A000312(n)*A000142(n)/A051696(n).

More terms from James Sellers, Jul 13 2000

cp's OEIS Frontend

A067911 Product of gcd(k,n) for 1 <= k <= n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

Extensions

A095996 a(n) = largest divisor of n! that is coprime to n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Formula

A064446 a(n) = gcd(n!, n^n, lcm(1, 2, ..., n)), or gcd(n^n, lcm(1, 2, ..., n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

PARI

PARI

Formula

A062763 a(n) is the greatest common divisor of (n-1)! and n^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A055774 Least common multiple of n! and n^n.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula

Extensions