A088973 - OEIS

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

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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: A034123 A338126 A360368 * 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