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

%I A271387 #15 Feb 16 2025 08:33:33
%S A271387 1,2,3,5,35,77,1001,2431,46189,1062347,30808063,86822723,3212440751,
%T A271387 10131543907,435656388001,20475850236047,1085220062510491,
%U A271387 3766351981654057,229747470880897477,810162134158954261,57521511525285752531,4199070341345859934763,331726556966322934846277
%N A271387 Numerator of prime(n)#/n!, where prime(n)# is the prime factorial function.
%H A271387 Ilya Gutkovskiy, <a href="/A271387/b271387.txt">Table of n, a(n) for n = 0..75</a>
%H A271387 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Primorial.html">Primorial</a>
%H A271387 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>
%H A271387 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Factorial.html">Factorial</a>
%F A271387 a(n) = prime(n)#/GCD(prime(n)#, n!), where GCD(a, b) is the greatest common divisor.
%F A271387 a(n) = prime(n)#/prime(pi(n))#, where pi(n) is the number of primes <= n.
%F A271387 a(n) = A002110(n)/A034386(n) = A002110(n)/A002110(A000720(n)) = A002110(n)/A007947(A000142(n)).
%e A271387 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, ...
%e A271387 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.
%t A271387 Table[Numerator[Product[Prime@ k, {k, n}]/n!], {n, 0, 22}] (* _Michael De Vlieger_, Apr 08 2016 *)
%o A271387 (PARI) a(n) = numerator(prod(k=1, n, prime(k))/n!); \\ _Michel Marcus_, Apr 09 2016
%Y A271387 Cf. A000040, A000142, A000720, A002110, A007947, A034386, A049614 (denominator of prime(n)#/n!), A090586, A135568.
%K A271387 nonn,frac
%O A271387 0,2
%A A271387 _Ilya Gutkovskiy_, Apr 06 2016