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A336374 Numbers k such that gcd(k, prime(k) + prime(k+2)) = 1.

%I A336374 #8 Apr 21 2021 03:47:30
%S A336374 1,3,5,7,11,13,15,17,19,23,27,29,31,35,37,39,41,43,47,49,53,55,59,61,
%T A336374 63,65,67,69,71,73,77,79,81,83,85,87,89,91,93,95,97,99,101,103,107,
%U A336374 109,113,115,119,121,127,129,131,135,137,139,141,143,147,149,151
%N A336374 Numbers k such that gcd(k, prime(k) + prime(k+2)) = 1.
%C A336374 This sequence and A336374 partition the positive integers.
%e A336374 In the following table, p(k) = A000040(k) = prime(k).
%e A336374   k    p(k)   p(k)+p(k+2)   gcd
%e A336374   1     2         7          1
%e A336374   2     3        10          2
%e A336374   3     5        16          1
%e A336374   4     7        20          4
%e A336374   5    11        28          1
%e A336374   6    13        32          2
%e A336374 1 and 3 are in this sequence; 2 and 4 are in A336375; 2 and 5 are in A336376; 3 and 7 are in A336377.
%t A336374 p[n_] := Prime[n];
%t A336374 u = Select[Range[200], GCD[#, p[#] + p[# + 2]] == 1 &]  (* A336374 *)
%t A336374 v = Select[Range[200], GCD[#, p[#] + p[# + 2]] > 1 &]   (* A336375 *)
%t A336374 Prime[u]  (* A336376 *)
%t A336374 Prime[v]  (* A336377 *)
%Y A336374 Cf. A000040, A336366, A336375, A336376, A336377.
%K A336374 nonn
%O A336374 1,2
%A A336374 _Clark Kimberling_, Oct 06 2020