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A336409 Distance from prime(n) to the nearest odd composite that is < prime(n).

%I A336409 #23 Oct 05 2020 02:41:43
%S A336409 2,4,2,4,2,2,4,2,2,4,2,2,2,4,2,2,4,2,2,2,2,2,4,2,4,2,2,2,2,4,2,4,2,2,
%T A336409 2,2,2,4,2,4,2,4,2,2,2,4,2,2,4,2,2,2,2,4,2,2,4,2,2,2,4,2,2,2,2,4,2,2,
%U A336409 2,2,2,2,2,2,2,2,2,4,2,4,2,2,2,2,2,4
%N A336409 Distance from prime(n) to the nearest odd composite that is < prime(n).
%F A336409 a(n) = 2 * A175191(n-1). - _Alois P. Heinz_, Oct 02 2020
%F A336409 a(n) = 2 * (A062301(n) + 1). - _Hugo Pfoertner_, Oct 02 2020
%e A336409 Beginning with prime(5) = 11:  11-9 = 2, 13-9 = 4, 17-15 = 2, 19-15 = 4.
%p A336409 A336409 := proc(n)
%p A336409     local p;
%p A336409     p := ithprime(n) ;
%p A336409     for a from p-2 by -2 do
%p A336409         if not isprime(a) then
%p A336409             return p-a ;
%p A336409         end if;
%p A336409     end do:
%p A336409 end proc:
%p A336409 seq(A336409(n),n=5..100) ; # _R. J. Mathar_, Oct 02 2020
%p A336409 # second Maple program:
%p A336409 a:= n-> `if`(isprime(ithprime(n)-2), 4, 2):
%p A336409 seq(a(n), n=5..100);  # _Alois P. Heinz_, Oct 02 2020
%t A336409 z = 5000; d = Select[Range[2, z], ! PrimeQ@# && OddQ@# &];  (* A014076 *)
%t A336409 f[n_] := Select[d, # < Prime[n] &];
%t A336409 t = Table[Prime[n] - Max[f[n]], {n, 5, 300}]  (* A336409 *)
%t A336409 Flatten[Position[t, 2]]  (* A336410 *)
%t A336409 Flatten[Position[t, 4]]  (* A336411 *)
%Y A336409 Cf. A175191, A014076, A336410, A336411, A062301.
%K A336409 nonn
%O A336409 5,1
%A A336409 _Clark Kimberling_, Sep 06 2020