cp's OEIS Frontend

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

%I A075541 #22 Jan 14 2025 14:58:03
%S A075541 2,15,36,39,46,54,55,73,96,99,102,107,110,118,129,160,164,167,179,184,
%T A075541 187,194,199,202,218,231,238,239,242,271,272,273,274,290,291,292,311,
%U A075541 326,339,356,357,358,362,387,419,426,437,438,449,452,464,465,489,508
%N A075541 Numbers k such that prime(k) + prime(k+1) + prime(k+2) is a multiple of 3.
%C A075541 Not every three successive primes have an integer average. The integer averages are in A075540.
%C A075541 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.
%H A075541 Robert Israel, <a href="/A075541/b075541.txt">Table of n, a(n) for n = 1..10000</a>
%e A075541 a(2) = 15 because (prime(15)+prime(16)+prime(17)) = (1/3)*(47 + 53 + 59) = 53 (integer average of three successive primes).
%p A075541 R:= NULL: count:= 0:
%p A075541 q:= 2: r:= 3:
%p A075541 for i from 1 while count < 100 do
%p A075541   p:= q; q:= r; r:= nextprime(r);
%p A075541   if p+q+r mod 3 = 0 then
%p A075541      R:= R,i; count:= count+1
%p A075541   fi
%p A075541 od:
%p A075541 R; # _Robert Israel_, Nov 10 2024
%t A075541 A075541= {}; Do[If[IntegerQ[s3 = (Prime[i] + Prime[i + 1] + Prime[i + 2])/3], A075541 = Append[A075541, i]], {i, 1000}]; (* 119 terms*)
%Y A075541 Cf. A006562, A024675, A075540, A064113.
%K A075541 nonn
%O A075541 1,1
%A A075541 _Zak Seidov_, Sep 21 2002