A194659 - OEIS

%I #25 Sep 23 2018 20:59:25

%S 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 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 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 a(n) = A104272(n) - A194658(n).

%C Conjecture 1. The sequence is unbounded.

%C 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 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 Conjecture 2. The asymptotic density of nonzero terms is 2/(e^2+1).

%H Antti Karttunen, <a href="/A194659/b194659.txt">Table of n, a(n) for n = 1..16385</a>

%o (PARI)

%o up_to = 65537;

%o 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 v104272 = A104272list(65537);

%o A104272(n) = v104272[n];

%o A080359(n) = {my(x = 1); while((primepi(x) - primepi(x\2)) != n, x++; ); x; }; \\ From A080359

%o A194658(n) = precprime(A080359(1+n)-1);

%o A194659(n) = (A104272(n) - A194658(n)); \\ _Antti Karttunen_, Sep 21 2018

%Y Cf. A104272, A194658, A194186, A164288.

%K nonn

%O 1,13

%A _Vladimir Shevelev_, Sep 01 2011