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Number of primes of the form p(n)#/p(k) - 1, where 1 <= k <= n.
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%I #4 Mar 30 2012 17:36:44

%S 0,1,1,2,2,4,2,3,4,1,0,3,3,3,3,5,2,0,3,3,5,2,2,1,5,4,2,1,2,0,0,1,1,2,

%T 3,1,2,3,1,2,1,3,0,3,4,0,4,1,0,1,3,0,2,2,5,1,2,1,5,1,1,2,1,1,3,6,3,2,

%U 4,4,0,1,2,2,4,1,4,2,1,1,2,1,2,2,2,2,2,4,2,2,0,4,3,2,2,4,1,0,0,2,2,3,4,4,3

%N Number of primes of the form p(n)#/p(k) - 1, where 1 <= k <= n.

%C p(k) is k-th prime; p(n)# is n-th primorial, A002110(n). For the larger n, these are only counts of highly probable primes. Of the first 500 terms, the maximum occurs twice, a(366)=a(432)=7; the mode is 2, occurring 142 times.

%H Rick Shepherd, <a href="/A135714/b135714.txt">Table of n, a(n) for n = 1..500</a>

%e a(6)=4 because p(6)#=A002110(6)=30030 and 30030/3-1=10009, 30030/7-1=4289, 30030/11-1=2729 and 30030/13-1=2309 are all prime and there are no other primes of this form.

%o (PARI) a(n)= p=prod(k=1,n,prime(k)); sum(k=1,n,isprime(p/prime(k)-1))

%Y Cf. A135715, A135716, A002110.

%K nonn

%O 1,4

%A _Rick L. Shepherd_, Nov 28 2007