A080359 -id:A080359 - cp's OEIS frontend

A164368 Primes p with the property: if q is the smallest prime > p/2, then a prime exists between p and 2q.

2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 109, 127, 137, 149, 151, 167, 179, 181, 191, 197, 227, 229, 233, 239, 241, 263, 269, 281, 283, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491, 503, 521, 569, 571, 587, 593, 599, 601, 607

Offset: 1

Vladimir Shevelev, Aug 14 2009

nonn

The Ramanujan primes possess the following property:
If p = prime(n) > 2, then all numbers (p+1)/2, (p+3)/2, ..., (prime(n+1)-1)/2 are composite.
The sequence equals all primes with this property, whether Ramanujan or not.
All Ramanujan primes A104272 are in the sequence.
Every lesser of twin primes (A001359), beginning with 11, is in the sequence. - Vladimir Shevelev, Aug 31 2009
109 is the first non-Ramanujan prime in this sequence.
A very simple sieve for the generation of the terms is the following: let p_0=1 and, for n>=1, p_n be the n-th prime. Consider consecutive intervals of the form (2p_n, 2p_{n+1}), n=0,1,2,... From every interval containing at least one prime we remove the last one. Then all remaining primes form the sequence. Let us demonstrate this sieve: For p_n=1,2,3,5,7,11,... consider intervals (2,4), (4,6), (6,10), (10,14), (14,22), (22,26), (26,34), ... . Removing from the set of all primes the last prime of each interval, i.e., 3,5,7,13,19,23,31,... we obtain 2,11,17,29, etc. - Vladimir Shevelev, Aug 30 2011
This sequence and A194598 are the mutually wrapping up sequences:
   A194598(1) <= a(1) <= A194598(2) <= a(2) <= ...
From Peter Munn, Oct 30 2017: (Start)
The sequence is the list of primes p = prime(k) such that there are no primes between prime(k)/2 and prime(k+1)/2. Changing "k" to "k-1" and therefore "k+1" to "k" produces a definition very similar to A164333's: it differs by prefixing an initial term 3. From this we get a(n+1) = prevprime(A164333(n)) = A151799(A164333(n)) for n >= 1.
The sequence is the list of primes that are not the largest prime less than 2*prime(k) for any k, so that - as a set - it is the complement relative to A000040 of the set of numbers in A059788.
{{2}, A166252, A166307} is a partition.
(End)

			2 is in the sequence, since then q=2, and there is a prime 3 between 2 and 4. - _N. J. A. Sloane_, Oct 15 2009

Alois P. Heinz, Table of n, a(n) for n = 1..20000
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, arXiv:0909.0715 [math.NT], 2009-2011.
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4
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 (see Section 5 Prime gaps).
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 (see Section 5 Prime gaps).

Cf. Ramanujan primes, A104272, and related sequences: A164288, A080359, A164294, A193507, A194184, A194186.
Cf. A059788, A164333, A194598.
A001359, A166252, A166307 are subsets.
Cf. A001262, A001567, A062568, A141232 (all relate to pseudoprimes to base 2).

a:= proc(n) option remember; local q, k, p;
      k:= nextprime(`if`(n=1, 1, a(n-1)));
      do q:= nextprime(floor(k/2));
         p:= nextprime(k);
         if p<2*q then break fi;
         k:= p
      od; k
    end:
seq(a(n), n=1..55);  # Alois P. Heinz, Aug 30 2011

Reap[Do[q=NextPrime[p/2]; If[PrimePi[2*q] != PrimePi[p], Sow[p]], {p, Prime[Range[100]]}]][[2, 1]]
(* Second program: *)
fQ[n_] := PrimePi[ 2NextPrime[n/2]] != PrimePi[n];
Select[ Prime@ Range@ 105, fQ]

is(n)=nextprime(n+1)<2*nextprime(n/2) && isprime(n) \\ Charles R Greathouse IV, Apr 24 2015

As a set, this sequence = A000040 \ A059788 = A000040 \ prevprime(2*A000040) = A000040 \ A151799(A005843(A000040)). - Peter Munn, Oct 30 2017

Definition clarified and simplified by Jonathan Sondow, Oct 25 2011

A056171 a(n) = pi(n) - pi(floor(n/2)), where pi is A000720.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 3, 3, 4, 4, 4, 3, 4, 4, 4, 3, 3, 3, 4, 4, 5, 5, 5, 4, 4, 4, 5, 4, 4, 4, 5, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 6, 7, 7, 8, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 9, 9, 9, 9, 9, 10, 10, 10, 9, 10, 10, 10, 9, 9, 9, 10, 10, 10, 10, 10, 9, 9, 9, 10, 10

Offset: 1

Labos Elemer, Jul 27 2000

Also, the number of unitary prime divisors of n!. A prime divisor of n is unitary iff its exponent is 1 in the prime power factorization of n. In general, gcd(p, n/p) = 1 or p. Here we count the cases when gcd(p, n/p) = 1.
A unitary prime divisor of n! is >= n/2, hence their number is pi(n) - pi(n/2). - Peter Luschny, Mar 13 2011
See also the references and links mentioned in A143227. - Jonathan Sondow, Aug 03 2008
From Robert G. Wilson v, Mar 20 2017: (Start)
First occurrence of k is at n = A080359(k).
The last occurrence of k is at n = A080360(k).
The number of times k appears is A080362(k). (End)
Lev Schnirelmann proved that for every n, a(n) > (1/log_2(n))*(n/3 - 4*sqrt(n)) - 1 - (3/2)*log_2(n). - Arkadiusz Wesolowski, Nov 03 2017

			10! = 2^8 * 3^2 * 5^2 * 7. The only unitary prime divisor is 7, so a(10) = 1.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 214.

Daniel Forgues, Table of n, a(n) for n=1..100000
Ethan Berkove and Michael Brilleslyper, Subgraphs of Coprime Graphs on Sets of Consecutive Integers, Integers, Vol. 22 (2022), #A47, see p. 8.

Cf. A000720, A001221, A034444, A048105, A048656, A048657, A056169, A056172.
Cf. A014085, A060715, A104272, A143223, A143224, A143225, A143226, A143227.
Cf. A080359, A080360, A080362.

A056171 := proc(x)
     numtheory[pi](x)-numtheory[pi](floor(x/2)) ;
end proc:
seq(A056171(n),n=1..130) ; # N. J. A. Sloane, Sep 01 2015
A056171 := n -> nops(select(isprime,[$iquo(n,2)+1..n])):
seq(A056171(i),i=1..98); # Peter Luschny, Mar 13 2011

s=0; Table[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; s, {k,100}]
Table[PrimePi[n]-PrimePi[Floor[n/2]],{n,100}] (* Harvey P. Dale, Sep 01 2015 *)

A056171=n->primepi(n)-primepi(n\2) \\ M. F. Hasler, Dec 31 2016

from sympy import primepi
[primepi(n) - primepi(n//2) for n in range(1,151)] # Indranil Ghosh, Mar 22 2017

[prime_pi(n)-prime_pi(floor(n/2)) for n in range(1,99)] # Stefano Spezia, Apr 22 2025

a(n) = A000720(n) - A056172(n). - Robert G. Wilson v, Apr 09 2017

a(n) = A056169(n!). - Amiram Eldar, Jul 24 2024

Definition simplified by N. J. A. Sloane, Sep 01 2015

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

Original entry on oeis.org

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

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

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

A060756 a(n) is the smallest number for which exactly n primes are bounded between a(n) and 2a(n) exclusively.

Original entry on oeis.org

1, 2, 4, 9, 10, 16, 22, 27, 34, 36, 40, 51, 52, 55, 57, 70, 82, 87, 91, 96, 99, 100, 120, 121, 126, 135, 136, 142, 147, 159, 175, 177, 187, 190, 205, 210, 216, 217, 220, 222, 232, 246, 250, 255, 262, 289, 297, 300, 301, 304, 309, 310, 324, 327, 330, 339, 342

Offset: 0

Lekraj Beedassy, Apr 23 2001

nonn

a(n) is the first occurrence of n in A060715.

			a(10)=40 since ten primes,namely,41,43,47,53,59,61,67,71,73,79,first arise between 40 and its double.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A060715, A059316, A080359.

import Data.List (elemIndex)
import Data.Maybe (mapMaybe)
a060756 n = a060756_list !! n
a060756_list = map (+ 1) $ mapMaybe (`elemIndex` a060715_list) [0..]
-- Reinhard Zumkeller, Jan 05 2012

More terms from Larry Reeves (larryr(AT)acm.org), Jun 05 2001

A194658 a(n) is the maximal prime, such that for all primes x<=a(n) the number of primes in (x/2,x) is less than n.

Original entry on oeis.org

2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 109, 137, 151, 167, 179, 181, 191, 197, 233, 239, 241, 263, 269, 281, 283, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491, 503, 521, 571, 587, 593, 599, 601, 607, 617, 643, 647, 653

Offset: 1

Vladimir Shevelev, Sep 01 2011

nonn

The next prime after a(n) is A080359(n+1).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
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.

Subsequence of A164368.

Cf. A080359, A193507, A104272.

b[1] = 2; b[n_] := b[n] = Module[{x = b[n-1]}, While[PrimePi[x] - PrimePi[ Quotient[x, 2]] != n, x++]; x];
a[n_] := NextPrime[b[n+1], -1];
Array[a, 100] (* Jean-François Alcover, Nov 11 2018 *)

A080359(n) <= a(n) <= A104272(n).

cp's OEIS Frontend

A164368 Primes p with the property: if q is the smallest prime > p/2, then a prime exists between p and 2q.

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A056171 a(n) = pi(n) - pi(floor(n/2)), where pi is A000720.

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

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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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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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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.

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A163961 First differences of A116533.

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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)].