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A071681
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Number of ways to represent the n-th prime as arithmetic mean of two other primes.
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17
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0, 0, 1, 1, 2, 2, 3, 1, 3, 3, 2, 4, 4, 4, 4, 5, 5, 3, 5, 7, 5, 4, 5, 6, 6, 8, 6, 7, 6, 6, 8, 8, 10, 6, 10, 8, 8, 6, 10, 8, 9, 7, 9, 11, 10, 6, 10, 11, 11, 8, 12, 10, 10, 14, 13, 14, 13, 9, 10, 13, 12, 12, 14, 16, 11, 13, 13, 14, 18, 13, 18, 14, 14, 17, 14, 16, 14, 16, 15, 16, 16, 17, 16, 16
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OFFSET
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1,5
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COMMENTS
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Conjecture: a(n)>0 for n>2.
The conjecture follows from a slightly strengthened version of Goldbach's conjecture: that every even number > 6 is the sum of two distinct primes. - T. D. Noe, Jan 10 2011 [Corrected by Barry Cherkas and Robert Israel, May 21 2015]
Number of primes q < prime(n), such that 2*prime(n) - q is prime. - Dmitry Kamenetsky, May 27 2023
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LINKS
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EXAMPLE
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a(7)=3 as prime(7) = 17 = (3+31)/2 = (5+29)/2 = (11+23)/2 and 2*17-p is not prime for the other primes p < 17: {2,7,13}.
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MATHEMATICA
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f[n_] := Block[{c = 0, k = PrimePi@n - 1}, While[k > 0, If[ PrimeQ[2n - Prime@k], c++ ]; k-- ]; c]; Table[ f@ Prime@n, {n, 84}] (* Robert G. Wilson v, Mar 22 2007 *)
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PROG
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(PARI) A071681(n)={s=2*prime(n); a=0; for(i=1, n-1, a=a+isprime(s-prime(i))); a}
(Haskell)
a071681 n = sum $ map a010051' $
takeWhile (> 0) $ map (2 * a000040 n -) $ drop n a000040_list
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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