A202188 -id:A202188

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A168425

Large Associated Ramanujan Prime, p_i.

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

FORMULA

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

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

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

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

(PARI) 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(s<n, L[s+1]=k+1)); v=apply(x->nextprime(x+1), L); v} \\ Bill McEachen, Jun 24 2023 (incorporates code from A104272)

CROSSREFS

Cf. A104272, A168421, A179196, A190874.

Cf. A202187, A202188, A234298.

KEYWORD

nonn

AUTHOR

John W. Nicholson, Nov 25 2009

STATUS

approved

A190874

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

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.

FORMULA

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).

EXAMPLE

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

MATHEMATICA

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]}];

R = R + 1;

PrimePi[R] // Differences (* Jean-François Alcover, Nov 11 2018, after T. D. Noe in A104272 *)

CROSSREFS

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

KEYWORD

nonn

AUTHOR

John W. Nicholson, May 22 2011

STATUS

approved

A174641

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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.

MATHEMATICA

nn=10000; t=Table[0, {nn}]; len=Prime[3*nn]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s<nn, t[[s+1]]=k], {k, len}]; t=t+1; t=Complement[Prime[Range[PrimePi[t[[-1]]]]], t]; ind=PrimePi[t]; d=Differences[ind]; cnt=0; n=1; Join[{2}, Reap[Do[If[d[[i]]==1, cnt++; If[cnt==n, Sow[t[[i-n+1]]]; n++], cnt=0], {i, Length[d]}]][[2, 1]]]

PROG

(Perl) 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

CROSSREFS

Cf. A104272 (Ramanujan primes), A174635 (non-Ramanujan primes).

Cf. A174602 (runs of Ramanujan primes).

Cf. A202187, A202188.

KEYWORD

nonn

AUTHOR

T. D. Noe, Nov 29 2010

STATUS

approved

A202187

Index of A190874 at record terms.

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..19.

EXAMPLE

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.

KEYWORD

nonn

AUTHOR

John W. Nicholson, Dec 14 2011

EXTENSIONS

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

STATUS

approved

A234298

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..26.

PROG

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

(Perl) 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

CROSSREFS

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).

KEYWORD

nonn

AUTHOR

John W. Nicholson, Dec 22 2013

EXTENSIONS

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

STATUS

approved

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