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A111183 a(n) = prime(x) - pi(x) where x is the least x such that (prime(x+1) - pi(x+1)) - (prime(x) - pi(x)) = n.

%I A111183 #18 Mar 19 2025 10:22:30
%S A111183 2,3,5,15,47,19,339,80,168,128,185,196,103,275,1771,1871,1028,498,
%T A111183 3004,851,3641,1087,11845,1613,5402,2404,3182,2889,5225,4190,5461,
%U A111183 10585,16958,1280,22444,9357,56241,30129,24857,19006,34461,15852,224417,15401
%N A111183 a(n) = prime(x) - pi(x) where x is the least x such that (prime(x+1) - pi(x+1)) - (prime(x) - pi(x)) = n.
%C A111183 Conjecture: a(n) exists for every n.
%H A111183 Robert Israel, <a href="/A111183/b111183.txt">Table of n, a(n) for n = 1..262</a>
%p A111183 N:= 100: # for a(1) .. a(N)
%p A111183 p:= 2: m:= 0: b:= 2:V:= Vector(N): count:= 0:
%p A111183 for x from 2 while count < N do
%p A111183   p:= nextprime(p);
%p A111183   if isprime(x) then m:= m+1 fi;
%p A111183   bp:= b; b:= p-m;
%p A111183   v:= b-bp;
%p A111183   if v >= 1 and v <= N and V[v] = 0 then V[v]:= bp; count:= count+1 fi
%p A111183 od:
%p A111183 convert(V,list); # _Robert Israel_, Mar 16 2025
%o A111183 (PARI) a(n) = { for(x=1, oo, my(y=prime(x)-primepi(x), z=prime(x+1)-primepi(x+1)); if(z-y == n,return(y)) ); }
%Y A111183 Cf. A000040, A000720, A111181.
%K A111183 nonn
%O A111183 1,1
%A A111183 _Cino Hilliard_, Oct 22 2005