A271387 Numerator of prime(n)#/n!, where prime(n)# is the prime factorial function.

1, 2, 3, 5, 35, 77, 1001, 2431, 46189, 1062347, 30808063, 86822723, 3212440751, 10131543907, 435656388001, 20475850236047, 1085220062510491, 3766351981654057, 229747470880897477, 810162134158954261, 57521511525285752531, 4199070341345859934763, 331726556966322934846277

Offset: 0

Author

Ilya Gutkovskiy, Apr 06 2016

Keywords

• nonn
• frac

Examples

			1, 2, 3, 5, 35/4, 77/4, 1001/24, 2431/24, 46189/192, 1062347/1728, 30808063/17280, 86822723/17280, 3212440751/207360, 10131543907/207360, 435656388001/2903040, ...
a(8) = 46189, because prime(8)#/8! = (2*3*5*7*11*13*17*19)/(1*2*3*4*5*6*7*8) = 46189/192.
		

Links

• Ilya Gutkovskiy, Table of n, a(n) for n = 0..75
• Eric Weisstein's World of Mathematics, Primorial
• Eric Weisstein's World of Mathematics, Prime Counting Function
• Eric Weisstein's World of Mathematics, Factorial

Crossrefs

Cf. A000040, A000142, A000720, A002110, A007947, A034386, A049614 (denominator of prime(n)#/n!), A090586, A135568.

Programs

Mathematica

Table[Numerator[Product[Prime@ k, {k, n}]/n!], {n, 0, 22}] (* Michael De Vlieger, Apr 08 2016 *)

PARI

a(n) = numerator(prod(k=1, n, prime(k))/n!); \\ Michel Marcus, Apr 09 2016

Formula

a(n) = prime(n)#/GCD(prime(n)#, n!), where GCD(a, b) is the greatest common divisor.

a(n) = prime(n)#/prime(pi(n))#, where pi(n) is the number of primes <= n.

a(n) = A002110(n)/A034386(n) = A002110(n)/A002110(A000720(n)) = A002110(n)/A007947(A000142(n)).