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

1,1

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.

Links

Table of n, a(n) for n=1..80.

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)].

Example

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).

Prog

(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

Crossrefs

Cf. A006450, A049090, A077800, A088973.

Sequence in context: A215751 A115888 A143814 * A153400 A289076 A372519

Adjacent sequences: A088968 A088969 A088970 * A088972 A088973 A088974

Keyword

nonn

Author

Cino Hilliard, Oct 30 2003

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

Status

approved