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A336366 Numbers k such that gcd(k, prime(k) + prime(k+1)) = 1.

%I A336366 #18 Apr 21 2021 03:49:30
%S A336366 1,5,7,9,11,13,17,19,21,23,25,29,31,37,39,41,43,47,49,51,53,59,61,67,
%T A336366 71,73,77,79,83,85,89,91,95,97,99,101,103,107,109,111,113,115,119,121,
%U A336366 123,125,127,129,131,133,137,139,145,149,151,155,157,161,163
%N A336366 Numbers k such that gcd(k, prime(k) + prime(k+1)) = 1.
%C A336366 This sequence and A336367 partition the positive integers.
%e A336366 In the following table, p(k) = A000040(k) = prime(k).
%e A336366   k    p(k)   p(k)+p(k+1)   gcd
%e A336366   1     2          5         1
%e A336366   2     3          8         4
%e A336366   3     5         12         3
%e A336366   4     7         18         2
%e A336366   5    11         24         1
%e A336366   6    13         30         6
%e A336366 Thus 1 and 5 are in this sequence; 2 and 3 are in A336367; 2 and 11 are in A336368; 3 and 5 are in A336369.
%t A336366 p[n_] := Prime[n];
%t A336366 u = Select[Range[200], GCD[#, p[#] + p[# + 1]] == 1 &] (* A336366 *)
%t A336366 v = Select[Range[200], GCD[#, p[#] + p[# + 1]] > 1 &]   (* A336367 *)
%t A336366 Prime[u] (* A336368 *)
%t A336366 Prime[v] (* A336369 *)
%o A336366 (PARI) isok(m) = gcd(m, prime(m)+prime(m+1)) == 1; \\ _Michel Marcus_, Jul 20 2020
%Y A336366 Cf. A000040, A336367, A336368, A336369.
%K A336366 nonn
%O A336366 1,2
%A A336366 _Clark Kimberling_, Jul 19 2020