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A088973
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Number of twin prime pairs between consecutive prime-indexed primes of order 4. The bounds are included in the calculation.
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2
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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
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1,1
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COMMENTS
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Conjecture: The interval [PIPS4(n), PIPS4(n+1)] always contains at least one twin prime pair. (This implies the Twin Prime Conjecture.)
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LINKS
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FORMULA
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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)].
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EXAMPLE
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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).
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PROG
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(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)
return len([i for i in range(PIP(n, 4), PIP(n+1, 4), 2) if (is_prime(i) and is_prime(i+2))])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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