Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #17 Mar 08 2024 09:01:49
%S 6,22,35,46,55,62,74,82,91,115,118,119,142,143,155,158,166,202,203,
%T 206,214,215,218,259,262,295,298,299,302,323,326,355,358,362,391,395,
%U 451,466,478,482,502,511,514,526,535,542,551,559,562,583,586,611,623,626
%N Semiprimes k such that k+3 is also a semiprime.
%C When a(n+1) = a(n) + 3 we have that a(n) is a semiprime such that a(n) and a(n)+3 and a(n) + 3 + 3 are all semiprimes, hence at least 3 semiprimes in arithmetic progression with common difference 3. This subsequence begins 115, 155. There cannot be 4 semiprimes in arithmetic progression with common difference 3, starting with k, because modulo 4 we have {k, k+3, k+6, k+9} == {k+0, k+3, k+2, k+1} and one of these must be divisible by 4, hence a nonsemiprime (eliminating k = 4 by inspection).
%H Harvey P. Dale, <a href="/A123017/b123017.txt">Table of n, a(n) for n = 1..1000</a>
%F {a(n)} = {k such that k is in A001358 and k+3 is in A001358}.
%e a(1) = 6 because 6 = 2 * 3 is semiprime and 6 + 3 = 9 = 3^2 is semiprime.
%e a(2) = 22 because 22 = 2 * 11 and 22 + 3 = 25 = 5^2.
%e a(3) = 35 because 35 = 5 * 7 and 35 + 3 = 38 = 2 * 19.
%e a(4) = 46 because 46 = 2 * 23 and 46 + 3 = 49 = 7^2.
%e a(5) = 55 because 55 = 5 * 11 and 55 + 3 = 58 = 2 * 29.
%t semiprimeQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range@ 670, semiprimeQ[ # ] && semiprimeQ[ # + 3] &] (* _Robert G. Wilson v_, Aug 31 2007 *)
%t SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,700}],{1,_,_,1}][[All, 1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Mar 03 2017 *)
%Y Cf. A001358, A056809, A070552, A092125, A092126, A092127, A092128, A092129.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Nov 04 2006
%E More terms from _Robert G. Wilson v_, Aug 31 2007