A080359 The smallest integer x > 0 such that the number of primes in (x/2, x] equals n.

2, 3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 113, 139, 157, 173, 181, 191, 193, 199, 239, 241, 251, 269, 271, 283, 293, 313, 349, 353, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 523, 577, 593, 599, 601, 607, 613, 619, 647, 653, 659

Offset: 1

Author

Labos Elemer, Feb 21 2003

Keywords

• nonn

Comments

a(n) is the same as: Smallest integer x > 0 such that the number of unitary-prime-divisors of x! equals n.

Let p_n be the n-th prime. If p_n>3 is in the sequence, then all integers (p_n-1)/2, (p_n-3)/2, ..., (p_(n-1)+1)/2 are composite numbers. - Vladimir Shevelev, Aug 12 2009

For n >= 3, denote by q(n) the prime which is the nearest from the left to a(n)/2. Then there exists a prime between 2q(n) and a(n). The converse, generally speaking, is not true; i.e., there exist primes that are outside the sequence, but possess such property (e.g., 131). - Vladimir Shevelev, Aug 14 2009

See sequence A164958 for a generalization. - Vladimir Shevelev, Sep 02 2009

a(n) is the n-th Labos prime.

Examples

			n=5: in 31! five unitary-prime-divisors appear (firstly): {17,19,23,29,31}, while other primes {2,3,5,7,11,13} are at least squared. Thus a(5)=31.
Consider a(9)=71. Then the nearest prime < 71/2 is q(9)=31, and between 2q(9) and a(9), i.e., between 62 and 71 there exists a prime (67). - _Vladimir Shevelev_, Aug 14 2009
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..4460 from Daniel Forgues)
• N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, arXiv:1108.0475 [math.NT], 2011.
• N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13.
• Ethan Berkove and Michael Brilleslyper, Subgraphs of Coprime Graphs on Sets of Consecutive Integers, Integers (2022) Vol. 22, #A47, see p. 8.
• Vladimir Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.
• Vladimir Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
• Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4.
• Jonathan Sondow, MathWorld: Ramanujan Prime
• Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.
• Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009), 630-635.

Crossrefs

Cf. A056171, A056169, A000720, A000142.

Cf. A104272 (Ramanujan primes).

Cf. A060756, A080360 (largest integer x with n primes in (x/2,x]).

Cf. A164554, A164288, A164333, A164294, A164372, A164371, A212493, A212541.

Programs

Mathematica

nn=1000; t=Table[0, {nn+1}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s<=nn && t[[s+1]]==0, t[[s+1]]=k], {k, Prime[3*nn]}]; Rest[t]
(* Second program: *)
a[1] = 2; a[n_] := a[n] = Module[{x = a[n-1]}, While[(PrimePi[x]-PrimePi[Quotient[x, 2]]) != n, x++ ]; x]; Array[a, 54] (* Jean-François Alcover, Sep 14 2018 *)

PARI

a(n) = {my(x = 1); while ((primepi(x) - primepi(x\2)) != n, x++;); x;} \\ Michel Marcus, Jan 15 2014

Sage

def A():
    i = 0; n = 1
    while True:
        p = prime_pi(i) - prime_pi(i//2)
        if p == n:
            yield i
            n += 1
        i += 1
A080359 = A()
[next(A080359) for n in range(54)] # Peter Luschny, Sep 03 2014

Formula

a(n) = Min{x; Pi[x]-Pi[x/2]=n} = Min{x; A056171(x)=n}=Min{x; A056169(n!)=n}; where Pi()=A000720().

a(n) <= A193507(n) (cf. A194186). - Vladimir Shevelev, Aug 18 2011

Extensions

Definition corrected by Jonathan Sondow, Aug 10 2008

Shrunk title and moved part of title to comments by John W. Nicholson, Sep 18 2011