A202187 -id:A202187 - cp's OEIS frontend

A168425 Large Associated Ramanujan Prime, p_i.

3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 131, 151, 157, 173, 181, 191, 229, 233, 239, 241, 251, 269, 271, 283, 311, 313, 349, 353, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 571, 577, 593, 599, 601, 607, 613, 643, 647, 653, 659, 661

Offset: 1

John W. Nicholson, Nov 25 2009

nonn

a(n) is the smallest prime on the right side of the Ramanujan Prime Corollary, 2*p_(i-n) > p_i, for i > k where k = pi(p_k) = pi(R_n) That is, p_k is the n-th Ramanujan Prime, R_n and the k-th prime.
a(n) = nextprime(R_n) = nextprime(p_k), where nextprime(x) is the next prime larger than x.
This is very useful in showing the number of primes in the range [p_k, 2*p_(i-n)] is greater than or equal to 1. By taking into account the size of the gaps between primes in [p_(i-n),p_k], one can see that the average prime gap is about log(p_k) using the following R_n / (2*n) ~ log(R_n).
Proof of Corollary: See Wikipedia link.
The number of primes until the next Ramanujan prime, R_(n+1), can be found in A190874.
Srinivasan's Lemma (2014): p_(k-n) < (p_k)/2 if R_n = p_k and n > 1. Proof: By the minimality of R_n, the interval ((p_k)/2,p_k] contains exactly n primes, so p_(k-n) < (p_k)/2. - Jonathan Sondow, May 10 2014
In spite of the name Large Associated Ramanujan Prime, a(n) is not a Ramanujan prime for many values of n. - Jonathan Sondow, May 10 2014

			For n=10, the n-th Ramanujan prime is A104272(n)= 97, the value of k = 25, so i is >= 26, i-n >= 16, the i-n prime is 53, and 2*53 = 106. This leaves the range [97, 106] for the 26th prime which is 101. In this example, 101 is the large associated Ramanujan prime.

T. D. Noe, Table of n, a(n) for n = 1..1000
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
Vladimir Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009) 630-635.
Jonathan Sondow, John W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.
Jonathan Sondow, Ramanujan Prime in MathWorld
Anitha Srinivasan, An upper bound for Ramanujan primes, Integers, 19 (2014), #A19.
Wikipedia, Ramanujan Prime

Cf. A104272, A168421, A179196, A190874.

Cf. A202187, A202188, A234298.

nn = 100; 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]}
];
RamanujanPrimes = R + 1;
Prime[PrimePi[#]+1]& /@ RamanujanPrimes (* Jean-François Alcover, Nov 03 2018, after T. D. Noe in A104272 *)

genit(n=100)={my(L=vector(n),s=0,k=1,z);for(k=1,prime(3*n)-1,if(ispseudoprime(k),s++);if(k%2==0&&ispseudoprime(k/2),s--);if(snextprime(x+1),L);v} \\ Bill McEachen, Jun 24 2023 (incorporates code from A104272)

use ntheory ":all"; say next_prime(nth_ramanujan_prime($)) for 1..100; # _Dana Jacobsen, Dec 25 2015

a(n) = prime(primepi(A104272(n)) + 1).

a(n) = A151800(A104272(n)). - Michel Marcus, Jun 27 2023

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

Original entry on oeis.org

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).

A174641 Smallest prime that begins a run of n consecutive primes that are not Ramanujan primes.

Original entry on oeis.org

3, 3, 3, 73, 191, 191, 509, 2539, 2539, 5279, 9901, 9901, 9901, 11593, 11593, 55343, 55343, 55343, 55343, 55343, 174929, 174929, 174929, 225977, 225977, 225977, 225977, 225977, 534889, 534889, 534889, 534889, 534889, 534889, 534889, 534889, 2492317, 2492317

Offset: 1

T. D. Noe, Nov 29 2010

nonn

The run of 10 consecutive non-Ramanujan primes was mentioned by Sondow.

Dana Jacobsen, Table of n, a(n) for n = 1..107 (first 67 terms from T. D. Noe)
J. Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.
J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009) 630-635.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.

Cf. A104272 (Ramanujan primes), A174635 (non-Ramanujan primes).
Cf. A174602 (runs of Ramanujan primes).
Cf. A202187, A202188.

nn=10000; t=Table[0, {nn}]; len=Prime[3*nn]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s

use ntheory ":all";
my($k, $max, $start, $end, $inc, $p, $q, $r, $pi)
   = (0, 0, 0, 10, 1e9, 0, 2, [], prime_iterator(3));
while (1) {
  if (!@$r) {
    ($start, $end) = ($end+1, $end+$inc);
    $r = ramanujan_primes($start, $end);
  }
  ($p, $q, $k) = ($q, shift(@$r), 0);
  # $k = prime_count($p+1,$q-1);
  $k++ while $pi->() < $q;
  say ++$max," ",next_prime($p)   while $k > $max;
}
# Dana Jacobsen, Jul 14 2016

A202188 Index of A168425 when A174641(n) = A168425(m); a(n) = m.

Original entry on oeis.org

1, 1, 1, 9, 18, 18, 42, 165, 165, 317, 559, 559, 559, 634, 634

Offset: 1

John W. Nicholson, Dec 14 2011

nonn

Same as A202187, but with repeats.

A202186 Record term value of A190874.

Original entry on oeis.org

4, 5, 7, 8, 10, 11, 14, 16, 21, 24, 29, 37, 40, 48, 50, 51, 65, 66, 68

Offset: 1

John W. Nicholson, Dec 13 2011

nonn

a(n) = A190874(n) at record term. For index of record term, see A202187.

Each term of A174641 corresponds with a term in A168425 such that if A174641(a(n) - 1) = A168425(m) then m of A168425 = n of A202187. Note that a(n) - 1 is the value of the index n of A174641.

			With n = 4, A202187(4)=42, and a(4) = 8. So, A190874(42)=8. However,
A174641(a(4)-1) = A174641(8-1) = A168425(A202187(4)) = A168425(42) = 509.

a(19) from 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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A168425 Large Associated Ramanujan Prime, p_i.

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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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A174641 Smallest prime that begins a run of n consecutive primes that are not Ramanujan primes.

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A202188 Index of A168425 when A174641(n) = A168425(m); a(n) = m.

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A202186 Record term value of A190874.

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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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