cp's OEIS Frontend

A190874 First differences of A179196, pi(R_(n+1)) - pi(R_n) where R_n is A104272(n).

%I A190874 #38 Feb 15 2025 03:07:36
%S A190874 4,2,3,3,2,2,2,1,5,1,2,3,4,1,3,2,1,7,1,1,1,1,3,1,3,3,1,5,1,3,1,5,1,1,
%T A190874 2,1,1,4,4,1,2,8,1,2,1,1,1,1,5,1,1,1,1,3,5,1,2,2,3,4,2,1,1,3,1,4,7,1,
%U A190874 1,2,3,3,2,1,1,3,1,1,1,2,1,2,1,5,2,3
%N A190874 First differences of A179196, pi(R_(n+1)) - pi(R_n) where R_n is A104272(n).
%C A190874 The count of primes of the interval (R_n,R_(n+1)] where R_n is A104272(n).
%C A190874 The sequence A182873 is the first difference of Ramanujan primes R_(n+1)- R_n. While each non-Ramanujan prime is bound by Ramanujan primes, the maximal non-Ramanujan prime gap is less than the maximal Ramanujan prime gap, A182873, and the ratio of a(n)/A182873(n) is the average gap size at R_n.
%C A190874 Record terms of n, a(n) are in A202186, A202187. Each record term value of a(n) - 1 is the index m of A168425(m). A202188 is the index of A168425 when A174641(n) = A168425(m), it has repeated values of A202187.
%C A190874 Starting at index n = A191228(A174602(m)) in this sequence, the first instance of a count of m - 1 consecutive 1's is seen.
%C A190874 Limit inferior of a(n) is positive, because there are infinitely many Ramanujan primes and each term of the sequence is >= 1.
%C A190874 Limit superior of a(n)/log(pi(R_n)) is positive infinity. Equivalently, there are infinitely many n > 0 such that pi(R_(n+1)) > pi(R_n) + t log(pi(R_n)), for every t > 0.
%C A190874 For all n > 3, a(n) < n.
%C A190874 a(n) = rho(n+1) - rho(n) using rho(x) as defined in Sondow, Nicholson, Noe.
%H A190874 T. D. Noe, <a href="/A190874/b190874.txt">Table of n, a(n) for n = 1..10000</a>
%H A190874 J. Sondow, J. W. Nicholson, and T. D. Noe, <a href="http://arxiv.org/abs/1105.2249"> Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
%F A190874 a(n) = pi(R_(n+1)) - pi(R_n) or
%F A190874 a(n) = A000720(A104272(n+1)) - A000720(A104272(n)).
%F A190874 a(n) = A179196(n+1) - A179196(n).
%e A190874 R(4) = 29, the fourth Ramanujan prime, the next Ramanujan prime is a(4) = 3 primes away or R(5) = 41.
%t A190874 nn = 100;
%t A190874 R = Table[0, {nn}]; s = 0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s<nn, R[[s+1]] = k], {k, Prime[3 nn]}];
%t A190874 R = R + 1;
%t A190874 PrimePi[R] // Differences (* _Jean-François Alcover_, Nov 11 2018, after _T. D. Noe_ in A104272 *)
%Y A190874 Cf. A179196, A104272, A000720, A168421, A168425, A182873, A202186, A202187, A202188, A174641, A191228, A174602.
%K A190874 nonn
%O A190874 1,1
%A A190874 _John W. Nicholson_, May 22 2011