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A071681 Number of ways to represent the n-th prime as arithmetic mean of two other primes. 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Conjecture: a(n)>0 for n>2.
a(A137700(n))=n and a(m)<>n for m < A137700(n), A000040(A137700(n))=A126204(n). - Reinhard Zumkeller, Feb 07 2008
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]
a(n) = A116619(n) + 1. - Reinhard Zumkeller, Mar 27 2015
Number of primes q < prime(n), such that 2*prime(n) - q is prime. - Dmitry Kamenetsky, May 27 2023
LINKS
Mladen Vassilev-Missana, Goldbach's n-perfect numbers as a key for proving the Goldbach's Conjecture, Notes on Number Theory and Discrete Mathematics (2005) Vol. 11, No. 1, 20-22.
EXAMPLE
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}.
MATHEMATICA
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 *)
PROG
(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
-- Reinhard Zumkeller, Mar 27 2015
CROSSREFS
Sequence in context: A335017 A047972 A004595 * A135621 A369366 A224764
KEYWORD
nonn,look
AUTHOR
Reinhard Zumkeller, May 31 2002
STATUS
approved

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Last modified March 19 06:53 EDT 2024. Contains 370953 sequences. (Running on oeis4.)