A075541 Indices of primes p(i) such that (1/3) (p(i)+p(i+1)+p(i+2)) is an integer.

2, 15, 36, 39, 46, 54, 55, 73, 96, 99, 102, 107, 110, 118, 129, 160, 164, 167, 179, 184, 187, 194, 199, 202, 218, 231, 238, 239, 242, 271, 272, 273, 274, 290, 291, 292, 311, 326, 339, 356, 357, 358, 362, 387, 419, 426, 437, 438, 449, 452, 464, 465, 489, 508

Offset

1,1

Comments

Not every three successive primes have an integer average. The integer averages are in A075540.

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 first 117 3-averages, there are 59 primes. Indices i(n) of first prime in sequence of three primes with integer average are in sequence A064113. Interprimes (s-averages with s=2) are all composite, see A024675.

Links

Table of n, a(n) for n=1..54.

Formula

i(n)-> 1/3 (p(i)+p(i+1)+p(i+2)) is integer.

Example

i(2) = 15 because (p(15)+p(16)+p(17)) = 1/3(47 + 53 + 59)=53 (integer average of three successive primes).

Mathematica

A075541= {}; Do[If[IntegerQ[s3 = (Prime[i] + Prime[i + 1] + Prime[i + 2])/3], A075541 = Append[A075541, i]], {i, 1000}]; (* 119 terms*)

Crossrefs

Cf. A006562, A024675, A075540, A064113.

Sequence in context: A301357 A271909 A346641 * A075542 A064113 A007217

Adjacent sequences: A075538 A075539 A075540 * A075542 A075543 A075544

Keyword

nonn

Author

Zak Seidov, Sep 21 2002

Status

approved