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A194659 a(n) = A104272(n) - A194658(n).

%I A194659 #25 Sep 23 2018 20:59:25
%S A194659 0,0,0,0,0,0,0,0,0,0,0,0,18,12,0,0,0,0,36,32,0,0,0,0,0,0,24,0,0,0,0,0,
%T A194659 0,0,0,0,0,0,0,0,0,0,48,0,0,0,0,0,0,24,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,
%U A194659 0,0,30,0,0,0,0,18,0,0,0,16,0,0,0,0,0,0,0,34,0,0,0,0,0,0,0,0,0,48,0,0,0,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,0,44,40
%N A194659 a(n) = A104272(n) - A194658(n).
%C A194659 Conjecture 1. The sequence is unbounded.
%C A194659 Records are 0, 18, 36, 48, 64, 84, 114, 138, 184, 202, 214, 268, 282, 366, 374, 378, 412, 444, 528, ... with indices 1, 13, 19, 43, 144, 145, 167, 560, 635, 981, 982, 2605, 3967, 4582, 7422, 7423, 7424, 7425, 10320, ... .
%C A194659 The places of nonzero terms correspond to places of those terms of A194658 which are in A164288. Moreover, for n>=1, places of nonzero terms of A194659 and A194186(n+1) coincide. This means that these sequences have the same lengths of the series of zeros.
%C A194659 Conjecture 2. The asymptotic density of nonzero terms is 2/(e^2+1).
%H A194659 Antti Karttunen, <a href="/A194659/b194659.txt">Table of n, a(n) for n = 1..16385</a>
%o A194659 (PARI)
%o A194659 up_to = 65537;
%o A194659 A104272list(n) = { my(L=vector(n), s=0, k=1); for(k=1, prime(3*n)-1, if(isprime(k), s++); if(k%2==0 && isprime(k/2), s--); if(s<n, L[s+1] = k+1)); (L); } \\ From A104272 by _Satish Bysany_, Mar 02 2017
%o A194659 v104272 = A104272list(65537);
%o A194659 A104272(n) = v104272[n];
%o A194659 A080359(n) = {my(x = 1); while((primepi(x) - primepi(x\2)) != n, x++; ); x; }; \\ From A080359
%o A194659 A194658(n) = precprime(A080359(1+n)-1);
%o A194659 A194659(n) = (A104272(n) - A194658(n)); \\ _Antti Karttunen_, Sep 21 2018
%Y A194659 Cf. A104272, A194658, A194186, A164288.
%K A194659 nonn
%O A194659 1,13
%A A194659 _Vladimir Shevelev_, Sep 01 2011