A202186 -id:A202186 - cp's OEIS frontend

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

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, 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, 1, 2, 3, 3, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 5, 2, 3

Offset: 1

John W. Nicholson, May 22 2011

nonn

The count of primes of the interval (R_n,R_(n+1)] where R_n is A104272(n).
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.
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.
Starting at index n = A191228(A174602(m)) in this sequence, the first instance of a count of m - 1 consecutive 1's is seen.
Limit inferior of a(n) is positive, because there are infinitely many Ramanujan primes and each term of the sequence is >= 1.
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.
For all n > 3, a(n) < n.
a(n) = rho(n+1) - rho(n) using rho(x) as defined in Sondow, Nicholson, Noe.

			R(4) = 29, the fourth Ramanujan prime, the next Ramanujan prime is a(4) = 3 primes away or R(5) = 41.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.

Cf. A179196, A104272, A000720, A168421, A168425, A182873, A202186, A202187, A202188, A174641, A191228, A174602.

nn = 100;
R = Table[0, {nn}]; s = 0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[sJean-François Alcover, Nov 11 2018, after T. D. Noe in A104272 *)

a(n) = pi(R_(n+1)) - pi(R_n) or
a(n) = A000720(A104272(n+1)) - A000720(A104272(n)).
a(n) = A179196(n+1) - A179196(n).

A202187 Index of A190874 at record terms.

Original entry on oeis.org

1, 9, 18, 42, 165, 317, 559, 634, 2604, 7421, 9401, 20760, 86376, 178008, 3549196, 3840921, 5633768, 16454380, 28751054

Offset: 1

John W. Nicholson, Dec 14 2011

nonn

Each term of A174641 corresponds with a term in A168425 such that if A174641(A202186(n) - 1) = A168425(m) then m of A168425 = n of a(n). Note that A202186(n) - 1 is the value of the index n of A174641.

Same as A202188, but without repeats.

			With n = 4, a(4)=42, and A202186(4) = 8. So, A190874(42)=8.
However, A174641(A202186(4)-1) = A174641(8-1) = A168425(a(4)) = A168425(42) = 509.

a(12)-a(19) added by John W. Nicholson, Jan 06 2014

A234298 Ramanujan prime R_k such that pi(R_(k+1)) - pi(R_k) are record values: record Ramanujan prime A190874(k).

Original entry on oeis.org

2, 71, 181, 503, 2531, 5273, 9887, 11587, 55339, 174917, 225961, 534883, 2492311, 5409337, 130449773, 141833603, 212583797, 658046911, 1183597123, 2897211971, 5602581277, 46992178547, 70637059291, 158465541049, 182591976709, 339683208863

Offset: 1

John W. Nicholson, Dec 22 2013

nonn

These are the primes preceding the unique values of A174641. That sequence is the start of a run of non-Ramanujan primes, so the previous prime is the Ramanujan prime. - Dana Jacobsen, Jul 14 2016

Record values are in A202186, index of A190874 at record terms in A202187, A202188 is the index of A168425 when A174641(n) = A168425(m); A202188(n) = m. A202187 is also the index of a(n).

perl -Mntheory=:all -nE 'my $n = $1 if /(\d+)$/; say ++$x," ",prev_prime($n) unless $seen{$n}++;' b174641.txt  # Dana Jacobsen, Jul 14 2016

use ntheory ":all"; my($max,$r)=(0,ramanujan_primes(1e7)); for (0..$#$r-1) { my $d=prime_count($r->[$+1])-prime_count($r->[$]); if ($d > $max) { say $r->[$]; $max=$d; } } # _Dana Jacobsen, Jul 14 2016

a(20) to a(26) from Dana Jacobsen, Jul 14 2016

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

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A202187 Index of A190874 at record terms.

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A234298 Ramanujan prime R_k such that pi(R_(k+1)) - pi(R_k) are record values: record Ramanujan prime A190874(k).

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