1,1

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. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), 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). Converse, generally speaking, is not true, i.e. there exist primes outside the sequence, but possess such property (e.g., 131) [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 14 2009]

A generalization. For k>1 (not necessarily integer), we call a Labos k-prime L_n^(k) the prime a_k(n) which is the smallest number such that pi(a_k(n)) - pi(a_k(n)/k)= n. Note that, the sequence of all primes corresponds to the case of "k=oo". Let p be a k-Labos prime, such that p/k is in the interval (p_m, p_(m+1)), where p_m>=3 and p_n is the nth prime. Then the interval (k*p_(m), p) contains a prime. Conjecture. For every k>1 there exist non-k-Labos primes, which possess the latter property. For example, for k=2, the smallest such prime is 131. Problem. For every k>1 to estimate the smallest non-k-Labos prime, which possess the latter property. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Sep 02 2009]

Daniel Forgues, Table of n, a(n) for n=1..4460

J. Sondow, Ramanujan Prime in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 10 2008]

V. Shevelev, On critical small intervals containing primes [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 20 2009]

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

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). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 14 2009]

Cf. A056171, A056169, A000720, A000142.

Cf. A104272 Ramanujan primes. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 10 2008]

A164554 A164288 A164333 A164294 A164372 A164371 [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 20 2009]

Sequence in context: A115898 A118134 A143871 this_sequence A103087 A135118 A084958

Adjacent sequences: A080356 A080357 A080358 this_sequence A080360 A080361 A080362

nonn

Labos E. (labos(AT)ana.sote.hu), Feb 21 2003

Definition corrected by Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 10 2008