cp's OEIS Frontend

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

%I A080359 #88 Feb 16 2025 08:32:48
%S A080359 2,3,13,19,31,43,53,61,71,73,101,103,109,113,139,157,173,181,191,193,
%T A080359 199,239,241,251,269,271,283,293,313,349,353,373,379,409,419,421,433,
%U A080359 439,443,463,491,499,509,523,577,593,599,601,607,613,619,647,653,659
%N A080359 The smallest integer x > 0 such that the number of primes in (x/2, x] equals n.
%C A080359 a(n) is the same as: Smallest integer x > 0 such that the number of unitary-prime-divisors of x! equals n.
%C A080359 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
%C A080359 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
%C A080359 See sequence A164958 for a generalization. - _Vladimir Shevelev_, Sep 02 2009
%C A080359 a(n) is the n-th Labos prime.
%H A080359 Amiram Eldar, <a href="/A080359/b080359.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..4460 from Daniel Forgues)
%H A080359 N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, <a href="http://arxiv.org/abs/1108.0475">Generalized Ramanujan primes</a>, arXiv:1108.0475 [math.NT], 2011.
%H A080359 N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, <a href="https://doi.org/10.1007/978-1-4939-1601-6_1">Generalized Ramanujan primes</a>, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13.
%H A080359 Ethan Berkove and Michael Brilleslyper, <a href="http://math.colgate.edu/~integers/w47/w47.pdf">Subgraphs of Coprime Graphs on Sets of Consecutive Integers</a>, Integers (2022) Vol. 22, #A47, see p. 8.
%H A080359 Vladimir Shevelev, <a href="http://arXiv.org/abs/0908.2319">On critical small intervals containing primes</a>, arXiv:0908.2319 [math.NT], 2009.
%H A080359 Vladimir Shevelev, <a href="http://arxiv.org/abs/0909.0715">Ramanujan and Labos primes, their generalizations and classifications of primes</a>, arXiv:0909.0715 [math.NT], 2009-2011.
%H A080359 Vladimir Shevelev, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos primes, their generalizations, and classifications of primes</a>, J. Integer Seq. 15 (2012) Article 12.5.4.
%H A080359 Jonathan Sondow, <a href="https://mathworld.wolfram.com/RamanujanPrime.html">MathWorld: Ramanujan Prime</a>
%H A080359 Jonathan Sondow, <a href="http://arxiv.org/abs/0907.5232">Ramanujan primes and Bertrand's postulate</a>, arXiv:0907.5232 [math.NT], 2009-2010.
%H A080359 Jonathan Sondow, <a href="http://www.jstor.org/stable/40391170">Ramanujan primes and Bertrand's postulate</a>, Amer. Math. Monthly, 116 (2009), 630-635.
%F A080359 a(n) = Min{x; Pi[x]-Pi[x/2]=n} = Min{x; A056171(x)=n}=Min{x; A056169(n!)=n}; where Pi()=A000720().
%F A080359 a(n) <= A193507(n) (cf. A194186). - _Vladimir Shevelev_, Aug 18 2011
%e A080359 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.
%e A080359 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
%t A080359 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]
%t A080359 (* Second program: *)
%t A080359 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 *)
%o A080359 (PARI) a(n) = {my(x = 1); while ((primepi(x) - primepi(x\2)) != n, x++;); x;} \\ _Michel Marcus_, Jan 15 2014
%o A080359 (Sage)
%o A080359 def A():
%o A080359     i = 0; n = 1
%o A080359     while True:
%o A080359         p = prime_pi(i) - prime_pi(i//2)
%o A080359         if p == n:
%o A080359             yield i
%o A080359             n += 1
%o A080359         i += 1
%o A080359 A080359 = A()
%o A080359 [next(A080359) for n in range(54)] # _Peter Luschny_, Sep 03 2014
%Y A080359 Cf. A056171, A056169, A000720, A000142.
%Y A080359 Cf. A104272 (Ramanujan primes).
%Y A080359 Cf. A060756, A080360 (largest integer x with n primes in (x/2,x]).
%Y A080359 Cf. A164554, A164288, A164333, A164294, A164372, A164371, A212493, A212541.
%K A080359 nonn
%O A080359 1,1
%A A080359 _Labos Elemer_, Feb 21 2003
%E A080359 Definition corrected by _Jonathan Sondow_, Aug 10 2008
%E A080359 Shrunk title and moved part of title to comments by _John W. Nicholson_, Sep 18 2011