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A075540
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Integers that are the average of three successive primes.
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9
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5, 53, 157, 173, 211, 257, 263, 373, 511, 537, 563, 593, 607, 653, 733, 947, 977, 999, 1073, 1103, 1123, 1187, 1223, 1239, 1367, 1461, 1501, 1511, 1541, 1747, 1753, 1763, 1773, 1899, 1907, 1917, 2071, 2181, 2287, 2401, 2409, 2417, 2449, 2677, 2903, 2963
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OFFSET
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1,1
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COMMENTS
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Not every three successive primes have an integer average. The integer averages are in the sequence.
Not all of these 3-averages are prime: the prime 3-averages are in A006562 (balanced primes). There are surprisingly many prime 3-averages: among the first 10000 terms of the sequence there are 2417 primes. Indices i(n) of first prime in sequence of three primes with integer average are in A075541, for prime 3-averages i(n) are in A064113. Interprimes (s-averages with s=2) are all composite, see A024675. (Edited by Zak Seidov, Sep 01 2015 )
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LINKS
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FORMULA
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a(n) = (1/3) (p(i)+p(i+1)+p(i+2)), for some i(n).
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EXAMPLE
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a(1) = 5 = (1/3)(3+5+7), first integer average of three successive primes; next is: a(2) = 53 = (1/3)(47 + 53 + 59); up to n=8 all terms are prime; while a(9) = 511 = (1/3)( 503 + 509 + 521) is the first nonprime 3-average: 511=7*73.
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MAPLE
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N:= 10^4: # to get all terms using primes <= N
Primes:= select(isprime, [2, seq(2*i+1, i=1..(N-1)/2)]):
select(type, (Primes[1..-3] + Primes[2..-2] + Primes[3..-1])/3, integer); # Robert Israel, Sep 01 2015
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MATHEMATICA
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Select[MovingAverage[Prime[Range[500]], 3], IntegerQ] (* Harvey P. Dale, Aug 10 2012 *)
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PROG
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(Haskell)
a075540 n = a075540_list !! (n-1)
a075540_list = map fst $ filter ((== 0) . snd) $
zipWith3 (\x y z -> divMod (x + y + z) 3)
a000040_list (tail a000040_list) (drop 2 a000040_list)
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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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Comment and example edited, inefficient Mma removed by Zak Seidov, Sep 01 2015
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STATUS
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approved
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