cp's OEIS Frontend

A088971 Number of twin prime pairs between consecutive prime-indexed primes of order 3. The bounds are included in the calculation.

%I A088971 #24 Mar 14 2020 13:38:48
%S A088971 3,5,8,12,9,16,12,15,33,16,32,19,12,23,27,31,7,54,24,14,32,30,33,54,
%T A088971 38,20,17,14,18,104,25,30,26,57,17,52,41,25,50,40,20,69,21,30,16,85,
%U A088971 135,18,18,22,28,28,65,26,63,64,17,45,29,15,93,115,41,13,21,129,56,80,17,25,31,59,70,70,37,33,41,42,58,92
%N A088971 Number of twin prime pairs between consecutive prime-indexed primes of order 3. The bounds are included in the calculation.
%C A088971 This sequence contains no 0's between x=1 to 33000. The interval [PIPS3(4133), PIPS3(4134)] contains 1 twin prime pair; the interval [PIPS3(8268), PIPS3(8269)] contains 2 twin prime pairs.
%F A088971 PIPS3(x) = A049090(x) = the x-th prime-indexed prime of order 3 = prime(prime(prime(prime(x)))) where prime(x) is the x-th prime. a(n) = count of twins in [PIPS3(n), PIPS3(n+1)].
%e A088971 a(1) = 3, since there are three pairs of twin primes at least PIPS3(1) = 11 and at most PIPS3(2) = 31: (11,13), (17,19), and (29,31).
%o A088971 (PARI) piptwins3(m,n) = { for(x=m,n, f=1; c=0; p1 = prime(prime(prime(prime(x)))); p2 = prime(prime(prime(prime(x+1)))); forprime(j=p1,p2-2, if(isprime(j+2),f=0; c++) ); print1(c","); ) }
%o A088971 (Sage)
%o A088971 def PIP(n, i): # Returns the n-th prime-indexed prime of order i.
%o A088971     if i==0:
%o A088971         return primes_first_n(n)[n-1]
%o A088971     else:
%o A088971         return PIP(PIP(n, i-1), 0)
%o A088971 def A088971(n): # Returns a(n)
%o A088971     return len([i for i in range(PIP(n, 3), PIP(n+1, 3), 2) if (is_prime(i) and is_prime(i+2))])
%o A088971 A088971(1) # _Danny Rorabaugh_, Mar 30 2015
%Y A088971 Cf. A006450, A049090, A077800, A088973.
%K A088971 nonn
%O A088971 1,1
%A A088971 _Cino Hilliard_, Oct 30 2003
%E A088971 Edited to count twin pairs entirely within [PIPS3(n), PIPS3(n+1)], rather than pairs with the first prime in that interval. - _Danny Rorabaugh_, Apr 01 2015