A075541 Numbers k such that prime(k) + prime(k+1) + prime(k+2) is a multiple of 3.

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

Robert Israel, Table of n, a(n) for n = 1..10000

Example

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

Maple

R:= NULL: count:= 0:
q:= 2: r:= 3:
for i from 1 while count < 100 do
  p:= q; q:= r; r:= nextprime(r);
  if p+q+r mod 3 = 0 then
     R:= R, i; count:= count+1
  fi
od:
R; # Robert Israel, Nov 10 2024

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