

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
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OFFSET

1,1


COMMENTS

Not every three successive primes have an integer average. The integer averages are in the sequence.
Not all of these 3averages are prime: the prime 3averages are in A006562 (balanced primes). There are surprisingly many prime 3averages: 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 3averages i(n) are in A064113. Interprimes (saverages 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 3average: 511=7*73.


MAPLE

N:= 10^4: # to get all terms using primes <= N
Primes:= select(isprime, [2, seq(2*i+1, i=1..(N1)/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 !! (n1)
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



