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A075540 Integers that are the average of three successive primes. 7
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 (list; graph; refs; listen; history; text; internal format)
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 a(n) are themselves 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

Cf. A006562, A024675, A075541, A064113.

Cf. A102655.

Sequence in context: A201017 A106097 A163580 * A006562 A094847 A001992

Adjacent sequences:  A075537 A075538 A075539 * A075541 A075542 A075543

KEYWORD

nonn

AUTHOR

Zak Seidov, Sep 21 2002

EXTENSIONS

Comment and example edited, inefficient Mma removed by Zak Seidov, Sep 01 2015

STATUS

approved

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Last modified February 23 06:42 EST 2020. Contains 332159 sequences. (Running on oeis4.)