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

5, 20, 25, 76, 51, 93, 61, 100, 176, 122, 207, 156, 89, 152, 249, 280, 44, 412, 178, 90, 293, 270, 282, 374, 340, 157, 186, 121, 169, 913, 263, 235, 255, 597, 162, 406, 457, 263, 418, 339, 221, 645, 161, 300, 133, 855, 1235, 236, 162, 240, 256, 243, 786, 261, 514, 590, 156, 481, 374, 211

Offset: 1

Author

Cino Hilliard, Oct 30 2003

Keywords

• nonn

Comments

Conjecture: The interval [PIPS4(n), PIPS4(n+1)] always contains at least one twin prime pair. (This implies the Twin Prime Conjecture.)

Examples

			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).
		

Crossrefs

Cf. A006450, A049203, A077800, A088971.

Programs

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","); ) }

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 A088973(n):
    return len([i for i in range(PIP(n,4),PIP(n+1,4),2) if (is_prime(i) and is_prime(i+2))])
A088973(60) # Danny Rorabaugh, Mar 30 2015

Formula

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)].

Extensions

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