A135715 - OEIS

%I #8 Nov 02 2017 09:15:42

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

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

%U 3,1,3,2,2,2,4,4,2,2,0,1,3,1,1,3,1,1,0,1,0,4,1,1,4,1,1,1,2,4,1,1,2,2,3,7,3

%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 once, a(172)=8; the mode is 2, occurring 135 times.

%H Rick L. Shepherd, <a href="/A135715/b135715.txt">Table of n, a(n) for n = 1..500</a>

%e a(3)=2 because p(3)#=A002110(3)=30 and 30/3+1=11 and 30/5+1=7 are both prime and there are no other primes of this form.

%t a[n_] := (p = Product[Prime[k], {k, 1, n}]; Sum[Boole[PrimeQ[p/Prime[k] + 1]], {k, 1, n}]); Array[a, 105] (* _Jean-François Alcover_, Nov 02 2017, translated from PARI *)

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

%Y Cf. A135714, A135716, A002110.

%K nonn

%O 1,3

%A _Rick L. Shepherd_, Nov 30 2007