A045375 - OEIS

%I #40 Jun 14 2026 10:11:30

%S 2,7,13,19,31,37,43,61,67,73,79,97,103,109,127,139,151,157,163,181,

%T 193,199,211,223,229,241,271,277,283,307,313,331,337,349,367,373,379,

%U 397,409,421,433,439,457,463,487,499,523,541,547,571,577,601,607,613,619,631,643,661,673,691

%N Primes congruent to {1, 2} mod 6.

%C Apart from initial term, same as A002476 = A007645 \ {2} = A045331 \ {2,3}. - _M. F. Hasler_, Apr 25 2008

%C Primes of the form 6*m - 3/2 -+ 5/2. A045375 UNION A045410 = A000040. - _Juri-Stepan Gerasimov_, Jan 28 2010

%C a(n)^k + 2 is composite for every positive integer k. Proof: For p = a(n) (i.e., p = 2 or p == 1 (mod 3)), p^k + 2 is composite for all k >= 1. If p = 2, then p^k + 2 = 2*(2^(k - 1) + 1) > 2. If p == 1 (mod 3), then p^k == 1 (mod 3), so p^k + 2 == 0 (mod 3) and > 3. - _Felix Huber_, May 27 2026

%H Felix Huber, <a href="/A045375/b045375.txt">Table of n, a(n) for n = 1..10000</a>

%p A045375List := proc(N)

%p     local i;

%p     [2, op(select(isprime, [seq(3*i + 1, i = 2 .. floor(N/3))]))];

%p end proc:

%p A045375List(691); # _Felix Huber_, May 27 2026

%t Select[Prime[Range[150]], MemberQ[{1, 2}, Mod[#, 6]] &] (* _Vladimir Joseph Stephan Orlovsky_, Feb 18 2012 *)

%o (Magma) [p: p in PrimesUpTo(740)|p mod 6 in {1,2}]; // _Vincenzo Librandi_, Dec 18 2010

%Y Cf. A002476, A007645, A045331.

%Y Cf. A000040 (the primes), A045410 (the primes of the form 6*k-2-+1). - _Juri-Stepan Gerasimov_, Jan 28 2010

%K nonn

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Vincenzo Librandi_, Dec 18 2010