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A088973 Number of twin prime pairs between consecutive prime-indexed primes of order 4. The bounds are included in the calculation. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..60.

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)].

EXAMPLE

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).

PROG

(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

CROSSREFS

Cf. A006450, A049203, A077800, A088971.

Sequence in context: A101728 A053240 A034123 * A005240 A147374 A080654

Adjacent sequences:  A088970 A088971 A088972 * A088974 A088975 A088976

KEYWORD

nonn

AUTHOR

Cino Hilliard, Oct 30 2003

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

STATUS

approved

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Last modified March 28 20:44 EDT 2020. Contains 333103 sequences. (Running on oeis4.)