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

3, 5, 8, 12, 9, 16, 12, 15, 33, 16, 32, 19, 12, 23, 27, 31, 7, 54, 24, 14, 32, 30, 33, 54, 38, 20, 17, 14, 18, 104, 25, 30, 26, 57, 17, 52, 41, 25, 50, 40, 20, 69, 21, 30, 16, 85, 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

Offset: 1

Author

Cino Hilliard, Oct 30 2003

Keywords

• nonn

Comments

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.

Examples

			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).
		

Crossrefs

Cf. A006450, A049090, A077800, A088973.

Programs

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","); ) }

Sage

def PIP(n, i): # Returns the n-th prime-indexed prime of order i.
    if i==0:
        return primes_first_n(n)[n-1]
    else:
        return PIP(PIP(n, i-1), 0)
def A088971(n): # Returns a(n)
    return len([i for i in range(PIP(n, 3), PIP(n+1, 3), 2) if (is_prime(i) and is_prime(i+2))])
A088971(1) # Danny Rorabaugh, Mar 30 2015

Formula

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)].

Extensions

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