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A038711 a(n) is the smallest m such that A002110(n) + m is prime.

%I A038711 #33 Dec 12 2023 10:31:39
%S A038711 1,1,1,1,1,1,17,19,23,37,61,1,61,71,47,107,59,61,109,89,103,79,151,
%T A038711 197,101,103,233,223,127,223,191,163,229,643,239,157,167,439,239,199,
%U A038711 191,199,383,233,751,313,773,607,313,383,293,443,331,283,277,271,401,307
%N A038711 a(n) is the smallest m such that A002110(n) + m is prime.
%C A038711 Any composite a(n) would disprove Fortune's conjecture, see A005235. - _Jeppe Stig Nielsen_, Oct 31 2003
%H A038711 Robert G. Wilson v, <a href="/A038711/b038711.txt">Table of n, a(n) for n = 0..1000</a>
%F A038711 a(n) = Min(1, A005235(n)); a(n)=1 for n=1, 2, 3, 4, 5, 11, 75, ...
%F A038711 a(n) = 1 for n=0, 1, 2, 3, 4, 5, 11, 75, ... (A014545); a(n) = A005235(n) otherwise. - _Jeppe Stig Nielsen_, Oct 31 2003
%F A038711 a(n) = A038710(n) - A002110(n). - _Alois P. Heinz_, Mar 16 2020
%e A038711 For n=11, 1 + A002110(11) = 200560490131 < 200560490197 = 67 + A002110(11); therefore, a(11)=1 but A005235(11)=67.
%p A038711 p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
%p A038711 a:= n-> nextprime(p(n))-p(n):
%p A038711 seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 16 2020
%t A038711 nmax=2^16384; npd=1;n=1;npd=npd*Prime[n]; While[npd<nmax, tt=1; cp=npd+tt; While[ !(PrimeQ[cp]), tt=tt+2;cp=cp+2]; Print[tt]; n=n+1; npd=npd*Prime[n]] (* _Lei Zhou_, Feb 15 2005 *)
%o A038711 (PARI) a(n) = my(P=vecprod(primes(n))); nextprime(P+1) - P; \\ _Michel Marcus_, Dec 12 2023
%Y A038711 Cf. A002110, A005235, A035345, A018239, A038710, A060270.
%K A038711 nonn
%O A038711 0,7
%A A038711 _Labos Elemer_, May 02 2000
%E A038711 a(0)=1 prepended by _Alois P. Heinz_, Mar 16 2020