cp's OEIS Frontend

A088973 Number of twin prime pairs between consecutive prime-indexed primes of order 4. The bounds are included in the calculation.

%I A088973 #24 Mar 14 2020 13:35:27
%S A088973 5,20,25,76,51,93,61,100,176,122,207,156,89,152,249,280,44,412,178,90,
%T A088973 293,270,282,374,340,157,186,121,169,913,263,235,255,597,162,406,457,
%U A088973 263,418,339,221,645,161,300,133,855,1235,236,162,240,256,243,786,261,514,590,156,481,374,211
%N A088973 Number of twin prime pairs between consecutive prime-indexed primes of order 4. The bounds are included in the calculation.
%C A088973 Conjecture: The interval [PIPS4(n), PIPS4(n+1)] always contains at least one twin prime pair. (This implies the Twin Prime Conjecture.)
%F A088973 PIPS4(x) = A049203(x) = the x-th prime-indexed primes of order 4 = prime(prime(prime(prime(prime(x))))) where prime(x) = A000040(x) is the x-th prime. a(n) = number of twin prime pairs in [PIPS4(n), PIPS(n+1)].
%e A088973 a(1) = 5, since there are five pairs of twin primes at least PIPS4(1) = 31 and at most PIPS4(2) = 127: (41,43), (59,61), (71,73), (101,103), and (107,109).
%o A088973 (PARI) piptwins4(m,n) = { for(x=m,n, f=1; c=0; p1 = prime(prime(prime(prime(prime(x))))); p2 = prime(prime(prime(prime(prime(x+1))))); forprime(j=p1,p2-2, if(isprime(j+2),f=0; c++) ); print1(c","); ) }
%o A088973 (Sage)
%o A088973 def PIP(n,i): # Returns the n-th prime-indexed prime of order i
%o A088973     if i==0:
%o A088973         return primes_first_n(n)[n-1]
%o A088973     else:
%o A088973         return PIP(PIP(n,i-1),0)
%o A088973 def A088973(n):
%o A088973     return len([i for i in range(PIP(n,4),PIP(n+1,4),2) if (is_prime(i) and is_prime(i+2))])
%o A088973 A088973(60) # _Danny Rorabaugh_, Mar 30 2015
%Y A088973 Cf. A006450, A049203, A077800, A088971.
%K A088973 nonn
%O A088973 1,1
%A A088973 _Cino Hilliard_, Oct 30 2003
%E A088973 Edited to count twin pairs entirely within [PIPS4(n), PIPS4(n+1)], rather than pairs with the first prime in that interval. - _Danny Rorabaugh_, Apr 01 2015