A104272 -id:A104272 - cp's OEIS frontend

A166251 Isolated primes: Primes p such that there is no other prime in the interval [2prevprime(p/2), 2nextprime(p/2)].

5, 7, 23, 37, 79, 83, 89, 163, 211, 223, 257, 277, 317, 331, 337, 359, 383, 389, 397, 449, 457, 467, 479, 541, 547, 557, 563, 631, 673, 701, 709, 761, 787, 797, 839, 863, 877, 887, 919, 929, 977, 1129, 1181, 1201, 1213, 1237, 1283, 1307, 1327, 1361, 1399, 1409

Offset: 1

Vladimir Shevelev, Oct 10 2009, Oct 14 2009

Other formulation:  Suppose a prime p >= 5 lies in the interval (2p_k, 2p_(k+1)), where p_n is the n-th prime; p is called isolated if the interval (2p_k, 2p_(k+1)) does not contain any other primes.
The sequence is connected with the following classification of primes: The first two primes 2,3 form a separate set of primes; let p >= 5 be in interval(2p_k, 2p_(k+1)), then 1)if in this interval there are primes only more than p, then p is called a right prime; 2) if in this interval there are primes only less than p, then p is called a left prime; 3) if in this interval there are prime more and less than p, then p is called a central prime; 4) if this interval does not contain other primes, then p is called an isolated prime. In particular, the right primes form sequence A166307 and all Ramanujan primes (A104272) more than 2 are either right or central primes; the left primes form sequence A182365 and all Labos primes (A080359) greater than 3 are either left or central primes.
From Peter Munn, Jun 01 2023: (Start)
The isolated primes are prime(k) such that k-1 and k occur as consecutive terms in A020900.
In the tree of primes described in A290183, the isolated primes label the nodes with no sibling nodes.
Conjecture: a(n)/A000040(n) is asymptotic to 9. This would follow from my conjectured asymptotic proportion of 1's in A102820 (the first differences of A020900).
(End)

			Since 2*17 < 37 < 2*19, and the interval (34, 38) does not contain other primes, 37 is an isolated prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4
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. A166307, A166252, A164368, A104272, A080359, A164333, A164288, A164294, A100484, A182426, A182365.

Cf. A020900, A102820, A290183.

a166251 n = a166251_list !! (n-1)
a166251_list = concat $ (filter ((== 1) . length)) $
   map (filter ((== 1) . a010051)) $
   zipWith enumFromTo a100484_list (tail a100484_list)
-- Reinhard Zumkeller, Apr 27 2012

isolatedQ[p_] := p == NextPrime[2*NextPrime[p/2, -1]] && p == NextPrime[2*NextPrime[p/2], -1]; Select[Prime /@ Range[300], isolatedQ] (* Jean-François Alcover, Nov 29 2012, after M. F. Hasler *)

is_A166251(n)={n==nextprime(2*precprime(n\2)) & n==precprime(2*nextprime(n/2))}  \\ M. F. Hasler, Oct 05 2012

Edited by N. J. A. Sloane, Oct 15 2009
More terms from Alois P. Heinz, Apr 26 2012
Given terms double-checked with new PARI code by M. F. Hasler, Oct 05 2012

A193507 Ramanujan primes of the second kind: a(n) is the smallest prime such that if prime x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.

Original entry on oeis.org

2, 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

Offset: 1

Vladimir Shevelev, Aug 18 2011

nonn

Apparently A168425 and the 2. - R. J. Mathar, Aug 25 2011
An odd prime p is in the sequence iff the previous prime is Ramanujan. The Ramanujan primes and the Ramanujan primes of the second kind are the mutually wrapping up sequences: a(1)<=R_1<=a(2)<=R_2<=a(3)<=R_3<=.... . - Vladimir Shevelev, Aug 29 2011
All terms of the sequence are in A194598. - Vladimir Shevelev, Aug 30 2011

			Since R_2=11 (see A104272), then for x >= 11, we have pi(x) - pi(x/2) >= 2. However, if to consider only prime x, then we see that, for x=7,5,3, pi(x) - pi(x/2)= 2, but pi(2) - pi(1)= 1. Therefore, already for prime x>=3, we have pi(x) - pi(x/2) >= 2. Thus a(2)=3.

V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
J. Sondow, Ramanujan Prime in MathWorld
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. A104272 (Ramanujan primes).

nn = 120; (* nn=120 returns 54 terms *)
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]}];
A104272 = R + 1;
Join[{2}, Select[Prime[Range[nn]], MemberQ[A104272, NextPrime[#, -1]]&]] (* Jean-François Alcover, Nov 07 2018, after T. D. Noe in A104272 *)

A080359(n) <= a(n) <= A104272(n) = R_n (Cf. A194184, A194186).

a(n)>p_(2*n-1); a(n)~p_{2n} (Cf. properties of R_n in A104272 and the above comment). - Vladimir Shevelev, Aug 28 2011

A194186 a(n) = A193507(n) - A080359(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 12, 0, 0, 0, 0, 36, 34, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 16, 0, 0, 0, 12, 0, 0, 0, 0, 0

Offset: 1

Vladimir Shevelev, Aug 18 2011

nonn

Conjecture: The sequence is unbounded.
Records are 0, 18, 36, 48, 52, ... with indices 1, 14, 20, 44, 96, ...
The places of nonzero terms correspond to places of those terms of A193507 which are in A164294. Conjecture: The asymptotic density of nonzero terms is 2/(e^2+1). A heuristic proof follows from the comment to A193507 and the first reference there.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A193507, A080359, A104272, A194184.

A164333 Primes prime(k) such that all integers in the interval [(prime(k-1)+1)/2, (prime(k)-1)/2] are composite numbers.

Original entry on oeis.org

13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 113, 131, 139, 151, 157, 173, 181, 191, 193, 199, 229, 233, 239, 241, 251, 269, 271, 283, 293, 311, 313, 349, 353, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 523, 571, 577, 593, 599, 601, 607, 613, 619, 643

Offset: 1

Vladimir Shevelev, Aug 13 2009

nonn

Let p_k be the k-th prime. A prime p is in the sequence iff the interval of the form (2p_k, 2p_(k+1)), containing p, also contains a prime less than p. The sequence is connected with the following classification of primes: the first two primes 2,3 form a separate set of primes; let p >= 5 be in the interval (2p_k, 2p_(k+1)), then 1) if in this interval there are only primes greater than p, then p is called a right prime; 2) if in this interval there are only primes less than p, then p is called a left prime; 3) if in this interval there are primes both greater and less than p, then p is called a central prime; 4) if this interval does not contain other primes, then p is called an isolated prime. In particular, the right primes form sequence A166307, and all Ramanujan primes (A104272) greater than 2 are either right or central primes; the left primes form sequence A182365, and all Labos primes (A080359) greater than 3 are either left or central primes; the central primes form A166252 and the isolated primes form A166251. [Vladimir Shevelev, Oct 10 2009] [Sequence reference updated by Peter Munn, Jun 01 2023]

Disjoint union of A166252 and A182365. - Peter Munn, Jun 01 2023 [an edited version of a contribution by Vladimir Shevelev in 2009]

			Let p=53. We see that 2*23<53<2*29. Since the interval (46, 58) contains prime 47<53 and does not contain any prime more than 53, then, by the considered classification 53 is left prime and it is in the sequence. [_Vladimir Shevelev_, Oct 10 2009]

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.
V. Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, J. Int. Seq. 15 (2012) # 12.5.4
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. A080359, A104272, A164288, A164294, A164332, A001262, A001567, A062568, A141232.
Cf. A164368, A164554, A166251, A166252. [Vladimir Shevelev, Oct 10 2009]
Cf. A166307, A182365.

isA164333 := proc(n)
        local i ;
        if isprime(n) and n > 3 then
                for i from (prevprime(n)+1)/2 to (n-1)/2 do
                        if isprime(i) then
                                return false;
                        end if;
                end do;
                return true;
        else
                false;
        end if;
end proc:
for i from 2 to 700 do
        if isA164333(i) then
                printf("%d,",i);
        end if;
end do: # R. J. Mathar, Oct 29 2011

kmax = 200; Select[Table[{(Prime[k - 1] + 1)/2, (Prime[k] - 1)/2}, {k, 3, kmax}], AllTrue[Range[#[[1]], #[[2]]], CompositeQ]&][[All, 2]]*2 + 1 (* Jean-François Alcover, Nov 14 2017 *)

{A080359} union {A164294} = {this sequence} union {2,3}. - Vladimir Shevelev, Oct 29 2011

A164368(2)A164368(3)A164368(4)Vladimir Shevelev, Oct 10 2009]

Definition rephrased by R. J. Mathar, Oct 02 2009

A116533 a(1)=1, a(2)=2, for n > 2 if a(n-1) is prime, then a(n) = 2*a(n-1), otherwise a(n) = a(n-1) - 1.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 9, 8, 7, 14, 13, 26, 25, 24, 23, 46, 45, 44, 43, 86, 85, 84, 83, 166, 165, 164, 163, 326, 325, 324, 323, 322, 321, 320, 319, 318, 317, 634, 633, 632, 631, 1262, 1261, 1260, 1259, 2518, 2517, 2516, 2515, 2514, 2513, 2512, 2511, 2510, 2509, 2508

Offset: 1

Rodolfo Kurchan, Mar 26 2006

nonn

For n >= 3, using Wilson's theorem, a(n) = a(n-1) + (-1)^r*gcd(a(n-1), W), where W = A038507(a(n-1) - 1), and r=1 if gcd(a(n-1), W) = 1 and r=0 otherwise. - Vladimir Shevelev, Aug 07 2009

Cf. A006992, A055496, A080359, A104272, A106108, A132199. - Vladimir Shevelev, Aug 07 2009

a[1]:=1: a[2]:=2: for n from 3 to 60 do if isprime(a[n-1])=true then a[n]:=2*a[n-1] else a[n]:=a[n-1]-1 fi od: seq(a[n],n=1..60); # Emeric Deutsch, Apr 02 2006

More terms from Emeric Deutsch, Apr 02 2006

A084140 a(n) is the smallest number j such that if x >= j there are at least n primes between x and 2x exclusively.

Original entry on oeis.org

2, 6, 9, 15, 21, 24, 30, 34, 36, 49, 51, 54, 64, 75, 76, 84, 90, 91, 114, 115, 117, 120, 121, 132, 135, 141, 154, 156, 174, 175, 184, 187, 201, 205, 210, 216, 217, 220, 231, 244, 246, 252, 285, 286, 294, 297, 300, 301, 304, 321, 322, 324, 327, 330, 339, 360, 364

Offset: 1

Harry J. Smith, May 15 2003

nonn

For all m >= a(n) there are at least n primes between m and 2m exclusively. This calculation relies on the fact that pi(2m) - pi(m) > m/(3*log(m)) for m >= 5. This is one more than the terms of A084139 with offset changed from 0 to 1.
For n > 5889, pi(2n) - pi(n) > f(2, 2n) - f(3, n) where f(k, x) = x/log x * (1 + 1/log x + k/(log x)^2). This may be useful for checking larger terms. The constant 3 can be improved at the cost of an increase in the constant 5889. - Charles R Greathouse IV, May 02 2012
A168421(n) = nextprime(a(n)), where nextprime(x) is the next prime >= x. - John W. Nicholson, Dec 21 2012
a(1) = ceiling((A104272(1)+1)/2) modifies the only even prime, 2; which has been stated, in Formula, as a(1) = A104272(1); for all others, a(n) = (A104272(n)+1)/2 = ceiling ((A104272(n)+1)/2). - John W. Nicholson, Dec 24 2012
Srinivasan's Lemma (2014): previousprime(a(n)) = p_(k-n) < (p_k)/2, where the n-th Ramanujan Prime R_n is the k-th prime p_k, and with 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. - Copied and adapted from a comment by Jonathan Sondow in A168421 by John W. Nicholson, Feb 17 2015

			a(11)=51 since there are at least 11 primes between m and 2m for all m >= 51 and this is not true for any m < 51. Although a(100)=720 is not listed, for all m >= 720, there are at least 100 primes between m and 2m.

Paulo Ribenboim, The Little Book of Big Primes, Springer-Verlag, 1991, p. 140.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag, 2004, p. 181.

Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv version, arXiv:1212.2785 [math.NT], 2012.
Anitha Srinivasan, An upper bound for Ramanujan primes, Integers, 19 (2014), #A19.
Eric Weisstein's World of Mathematics, Bertrand's Postulate.

Cf. A108954, A060715, A060756, A084138, A084139, A084141, A084142, A168421, A168425.

nn = 60;
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]}];
A104272 = R + 1;
Ceiling[(A104272 + 1)/2] (* Jean-François Alcover, Nov 07 2018, after T. D. Noe in A104272 *)

a(1) = A104272(1); for n >= 2, a(n) = (A104272(n)+1)/2. - Vladimir Shevelev, Dec 07 2012

a(n) = ceiling((A104272(n)+1)/2) for n >= 1. - John W. Nicholson, Dec 24 2012

A163961 First differences of A116533.

Original entry on oeis.org

1, 2, -1, 3, -1, 5, -1, -1, -1, 7, -1, 13, -1, -1, -1, 23, -1, -1, -1, 43, -1, -1, -1, 83, -1, -1, -1, 163, -1, -1, -1, -1, -1, -1, -1, -1, -1, 317, -1, -1, -1, 631, -1, -1, -1, 1259, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 2503, -1, -1, -1, 5003, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 1

Vladimir Shevelev, Aug 07 2009, Aug 14 2009

sign

Ignoring the +-1 terms, we obtain the sequence of Bertrand's primes A006992. If we consider sequences A_i={a_i(n)}, i=1,2,... with the same constructions as A116533, but with initials a_1(1)=2, a_2(1)=11, a_3(1)=17,..., a_m(1)=A164368(m),..., then the union of A_1,A_2,... contains all primes.

Cf. A116533, A006992, A055496, A080359, A104272, A106108, A132199, A164368

A116533 := proc(n) option remember; if n <=2 then n; else if isprime(procname(n-1)) then 2*procname(n-1) ; else procname(n-1)-1 ; end if; end if; end proc:
A163961 := proc(n) A116533(n+1)-A116533(n) ; end proc: # R. J. Mathar, Sep 03 2011

Differences@ Prepend[NestList[If[PrimeQ@ #, 2 #, # - 1] &, 2, 90], 1] (* Michael De Vlieger, Dec 06 2018 *)

a116533(n) = if(n==1, 1, if(n==2, 2, if(ispseudoprime(a116533(n-1)), 2*a116533(n-1), a116533(n-1)-1)))
a(n) = a116533(n+1)-a116533(n) \\ Felix Fröhlich, Dec 06 2018

lista(nn) = {va = vector(nn); va[1] = 1; va[2] = 2; for (n=3, nn, va[n] = if (isprime(va[n-1]), 2*va[n-1], va[n-1]-1);); vector(nn-1, n, va[n+1] - va[n]);} \\ Michel Marcus, Dec 07 2018

A163963 First differences of A080735.

Original entry on oeis.org

1, 2, 1, 5, 1, 11, 1, 23, 1, 47, 1, 1, 1, 97, 1, 1, 1, 197, 1, 1, 1, 397, 1, 1, 1, 797, 1, 1, 1, 1597, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3203, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6421, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12853, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25717, 1, 1, 1, 51437, 1, 1, 1

Offset: 1

Vladimir Shevelev, Aug 07 2009

nonn

Ignoring the 1 terms we obtain A055496. If we consider sequences A_i={a_i(n)}, i=1,2,... with the same constructions as A080735, but with initials a_1(1)=2, a_2(1)=3, a_3(1)=13,..., a_m(1)=A080359(m),..., then the union of A_1,A_2,... contains all primes.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A116533, A163961, A080735, A006992, A055496, A080359, A104272, A106108, A132199.

A080735 := proc(n) option remember; local p ; if n = 1 then 1; else p := procname(n-1) ; if isprime(p) then 2*p; else p+1 ; end if; end if; end proc: A163963 := proc(n) A080735(n+1)-A080735(n) ; end: seq(A163963(n),n=1..100) ; # R. J. Mathar, Nov 05 2009

Differences@ NestList[If[PrimeQ@ #, 2 #, # + 1] &, 1, 87] (* Michael De Vlieger, Dec 06 2018, after Harvey P. Dale at A080735 *)

lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = if (isprime(va[n-1]), 2*va[n-1], va[n-1]+1);); vector(nn-1, n, va[n+1] - va[n]);} \\ Michel Marcus, Dec 06 2018

More terms from R. J. Mathar, Nov 05 2009

A168421 Small Associated Ramanujan Prime, p_(i-n).

Original entry on oeis.org

2, 7, 11, 17, 23, 29, 31, 37, 37, 53, 53, 59, 67, 79, 79, 89, 97, 97, 127, 127, 127, 127, 127, 137, 137, 149, 157, 157, 179, 179, 191, 191, 211, 211, 211, 223, 223, 223, 233, 251, 251, 257, 293, 293, 307, 307, 307, 307, 307, 331, 331, 331

Offset: 1

John W. Nicholson, Nov 25 2009

nonn

a(n) is the smallest prime p_(k+1-n) on the left side of the Ramanujan Prime Corollary, 2*p_(i-n) > p_i for i > k, where the n-th Ramanujan Prime R_n is the k-th prime p_k. [Comment clarified and shortened by Jonathan Sondow, Dec 20 2013]
Smallest prime number, a(n), such that if x >= a(n), then there are at least n primes between x and 2x exclusively.
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.
Not the same as A124136.
A084140(n) is the smallest integer where ceiling ((A104272(n)+1)/2), a(n) is the next prime after A084140(n). - John W. Nicholson, Oct 09 2013
If a(n) is in A005382(k) then A005383(k) is a twin prime with the Ramanujan prime, A104272(n) = A005383(k) - 2, and A005383(k) = A168425(n). If this sequence has an infinite number of terms in A005382, then the twin prime conjecture can be proved. - John W. Nicholson, Dec 05 2013
Except for A000101(1)=3 and A000101(2)=5, A000101(k) = a(n). Because of the large size of a gap, there are many repeats of the prime number in this sequence. - John W. Nicholson, Dec 10 2013
For some n and k, we see that a(n) = A104272(k) as to form a chain of primes similar to a Cunningham chain. For example (and the first example), a(2) = 7, links A104272(2) = 11 = a(3), links A104272(3) = 17 = a(4), links A104272(4) = 29 = a(6), links A104272(6) = 47. Note that the links do not have to be of a form like q = 2*p+1 or q = 2*p-1. - John W. Nicholson, Dec 14 2013
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 Small Associated Ramanujan Prime, a(n) is not a Ramanujan prime for many values of n. - Jonathan Sondow, May 10 2014
Prime index of a(n), pi(a(n)) = i-n, is equal to A179196(n) - n + 1. - John W. Nicholson, Sep 15 2015
All maximal prime pairs in A002386 and A000101 are bounded by, for a particular n and i, the prime A104272(n) and twice a prime in A000040() following a(n). This means the gap between maximal prime pair cannot be more than twice the prior maximal prime gap. - John W. Nicholson, Feb 07 2019

			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, 53 is the small associated Ramanujan prime.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010; 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. Integer Seq. 14 (2011) Article 11.6.2.
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
Anitha Srinivasan, An upper bound for Ramanujan primes, Integers, 19 (2014), #A19
Wikipedia, Ramanujan Prime

Cf. A000101, A005382, A005383, A084139, A084140.
Cf. A104272, A124136, A168425, A179196, A190874.
Cf. A165959 (range size), A230147 (records).

nn = 100;
t = Table[0, {nn}];
Do[m = PrimePi[2n] - PrimePi[n]; If[0 < m <= nn, t[[m]] = n], {n, 15 nn}];
A084139 = Join[{1}, t];
a[n_] := NextPrime[A084139[[n]]];
Array[a, nn] (* Jean-François Alcover, Nov 07 2018, after T. D. Noe in A084139 *)

use ntheory ":all"; say next_prime((nth_ramanujan_prime($)+1) >> 1) for 1..100; # _Dana Jacobsen, Mar 02 2016

a(n) = prime(primepi(A104272(n)) + 1 - n).
a(n) = nextprime(A084139(n+1)), where nextprime(x) is the next prime > x. Note: some A084139(n) may be prime, therefore nextprime(x) not equal to x. - John W. Nicholson, Oct 11 2013
a(n) = nextprime(A084140(n)). - John W. Nicholson, Oct 11 2013

Extended by T. D. Noe, Nov 22 2010

A143227 (Number of primes between n and 2n) - (number of primes between n^2 and (n+1)^2), if > 0.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 2, 3, 3, 3, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 2, 6, 3, 3, 1, 1, 1, 2, 1, 1, 1, 1, 6, 3, 8, 3, 2, 3, 2, 3, 1, 1, 4, 3, 10, 2, 1, 1, 2, 3, 1, 3, 4, 2, 2, 9, 7, 2, 2, 4, 3, 3, 1, 2, 3, 5, 1, 2, 3, 2, 11, 3, 1, 2, 4, 7, 1, 1, 1, 1, 1, 5, 1, 2, 3, 3, 4, 2, 2, 9, 5, 1, 4, 2, 2

Offset: 1

Jonathan Sondow, Aug 02 2008

nonn

If the sequence is bounded (e.g., if it is finite), then Legendre's conjecture is true: there is always a prime between n^2 and (n+1)^2, at least for all sufficiently large n. This follows from the strong form of Bertrand's postulate proved by Ramanujan (see A104272 Ramanujan primes).

			The first positive value of ((pi(2n) - pi(n)) - (pi((n+1)^2) - pi(n^2))) is 1 (at n = 42), the 2nd is 2 (at n = 55) and the 3rd is 1 (at n = 56), so a(1) = 1, a(2) = 2, a(3) = 1.

M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1989, p. 19.
S. Ramanujan, Collected Papers of Srinivasa Ramanujan (G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson, eds.), Amer. Math. Soc., Providence, 2000, pp. 208-209.

T. D. Noe, Table of n, a(n) for n=1..413
T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.
M. Hassani, Counting primes in the interval (n^2,(n+1)^2), arXiv:math/0607096 [math.NT], 2006.
T. D. Noe, Plot of the points (A143226(n), A143227(n))
J. Pintz, Landau's problems on primes
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
J. Sondow, Ramanujan Prime in MathWorld.
J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld.
E. W. Weisstein, Legendre's Conjecture in MathWorld.

Cf. A000720, A014085, A060715, A104272, A143223, A143224, A143225, A143226 = corresponding values of n.

L={}; Do[ With[ {d=(PrimePi[2n]-PrimePi[n])-(PrimePi[(n+1)^2]-PrimePi[n^2])}, If[d>0, L=Append[L,d]]], {n,0,1000}]; L
Select[Table[(PrimePi[2n]-PrimePi[n])-(PrimePi[(n+1)^2]-PrimePi[n^2]),{n,1000}],#>0&] (* Harvey P. Dale, Jun 19 2019 *)

a(n) = |A143223(A143226(n))|.

cp's OEIS Frontend

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A193507 Ramanujan primes of the second kind: a(n) is the smallest prime such that if prime x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.

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A194186 a(n) = A193507(n) - A080359(n).

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A164333 Primes prime(k) such that all integers in the interval [(prime(k-1)+1)/2, (prime(k)-1)/2] are composite numbers.

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A084140 a(n) is the smallest number j such that if x >= j there are at least n primes between x and 2x exclusively.

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A163961 First differences of A116533.

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A163963 First differences of A080735.

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A166251 Isolated primes: Primes p such that there is no other prime in the interval [2prevprime(p/2), 2nextprime(p/2)].