A049614 n! divided by its squarefree kernel.

1, 1, 1, 1, 4, 4, 24, 24, 192, 1728, 17280, 17280, 207360, 207360, 2903040, 43545600, 696729600, 696729600, 12541132800, 12541132800, 250822656000, 5267275776000, 115880067072000, 115880067072000, 2781121609728000, 69528040243200000, 1807729046323200000

Offset: 0

Author

Labos Elemer

Keywords

• nonn

Comments

Also product of composite numbers less than or equal to n. - Benoit Cloitre, Aug 18 2002

Also n! divided by n primorial (or n!/n#). - Cino Hilliard, Mar 26 2006

From Alexander R. Povolotsky and Peter J. C. Moses, Aug 27 2007: (Start)

It appears that a(n) = smallest positive number m such that the sequence b(n) = { m (i^1 + 1!) (i^2 + 2!) ... (i^n + n!) / n! : i >= 0 } takes integral values. [It would be nice to have a proof of this! - N. J. A. Sloane] Cf. A064808 (for n=2), A131682 (for n=3), A131683 (for n=4), A131527 (for n=5), A131684 (for n=6), A131528. See also A129995, A131685. (End)

It appears that every term > 4 is divisible by 24. - Alexander R. Povolotsky, Oct 18 2007

The above comment is correct since each term divides the next. - Charles R Greathouse IV, Jan 16 2012

When n is not a prime number, then a(n)=m*n, where m is some integer >0; such a(n) make up the A036691 Otherwise, when n is a prime number, then a(n)=a(k), where k is the largest nonprime number preceding n (kAlexander R. Povolotsky, Aug 21 2012

Examples

			n = 11: 11! = 39916800 = 2310*17280 and 2310=2*3*5*7*11.
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• Index to divisibility sequences
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A000142, A002110, A003418, A034386, A036691, A045948, A048148.

Programs

Magma

A049614:= func< n | n le 1 select 1 else Factorial(n)/(&*[NthPrime(j): j in [1..#PrimesUpTo(n)]]) >;
[A049614(n): n in [0..40]]; // G. C. Greubel, Jul 21 2023

Maple

primorial := n -> mul(k, k=select(isprime, [$1..n]));
A049614 := n -> factorial(n)/primorial(n);
seq(A049614(i),i=0..24); # Peter Luschny, Feb 16 2013

Mathematica

Table[n!/Product[ Prime[i], {i, PrimePi[n]}], {n, 24}]

PARI

a(n)=prod(i=1,n,i^if(isprime(i),0,1))

PARI

a(n)=n!/prod(i=1,primepi(n),prime(i)) \\ Charles R Greathouse IV, Aug 30 2012

SageMath

def A049614(n): return factorial(n)/product(nth_prime(j) for j in range(1,1+prime_pi(n)))
[A049614(n) for n in range(41)] # G. C. Greubel, Jul 21 2023

Formula

a(n) = A000142(n)/A034386(n).

Extensions

Edited by N. J. A. Sloane, Oct 07 2007

Offset set to 0, a(0)=1 prepended to data, Peter Luschny, Feb 16 2013