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A083005 a(1)=1, a(n)=2a(n-1)+1 if a(n-1) is prime, a(n)=a(n-1)+1 otherwise.

%I A083005 #13 Jul 02 2021 13:08:42
%S A083005 1,2,5,11,23,47,95,96,97,195,196,197,395,396,397,795,796,797,1595,
%T A083005 1596,1597,3195,3196,3197,3198,3199,3200,3201,3202,3203,6407,6408,
%U A083005 6409,6410,6411,6412,6413,6414,6415,6416,6417,6418,6419,6420,6421,12843,12844
%N A083005 a(1)=1, a(n)=2a(n-1)+1 if a(n-1) is prime, a(n)=a(n-1)+1 otherwise.
%C A083005 Primes in this sequence are A055496.- _Robert Israel_, May 23 2017
%H A083005 Robert Israel, <a href="/A083005/b083005.txt">Table of n, a(n) for n = 1..10197</a>
%F A083005 Conjecture : limit n ->oo log(a(n))/sqrt(n)=c= 1.3....
%p A083005 P[0]:= 0:
%p A083005 for n from 2 to 20 do P[n]:= nextprime(2*P[n-1]) od:
%p A083005 seq($2*P[i]+1..P[i+1],i=0..19); # _Robert Israel_, May 23 2017
%t A083005 NestList[If[PrimeQ[#],2#+1,#+1]&,1,50] (* _Harvey P. Dale_, Jul 02 2021 *)
%Y A083005 Cf. A055496.
%K A083005 nonn
%O A083005 1,2
%A A083005 _Benoit Cloitre_, May 31 2003
%E A083005 Offset corrected by _Robert Israel_, May 23 2017