OFFSET
1,1
COMMENTS
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 )
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = (1/3) (p(i)+p(i+1)+p(i+2)), for some i(n).
EXAMPLE
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.
MAPLE
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
MATHEMATICA
Select[MovingAverage[Prime[Range[500]], 3], IntegerQ] (* Harvey P. Dale, Aug 10 2012 *)
PROG
(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)
-- Reinhard Zumkeller, Jan 20 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Zak Seidov, Sep 21 2002
EXTENSIONS
Comment and example edited, inefficient Mma removed by Zak Seidov, Sep 01 2015
STATUS
approved