OFFSET
1,3
COMMENTS
If the limit of R(n) exists as n->oo it is 1/2, but existence of the limit is conjectural. R(n) generalizes to R_k(n) by substituting PrimePi_k for PrimePi(n), where PrimePi_k(n) is the number of numbers with k prime factors (including repetitions) <= n. Convergence of {R(n)} to 1/2 implies Legendre's conjecture. For discussion of the order of the number of prime factors of a number n see reference [1], below. The PNT and reference [1] suggest but offer no proof that R_k(n)-> 1/2 as n -> oo. The corresponding sequence for near-primes would be {R_2(n)} = {1/3, 2/3, 1/2, ...}.
REFERENCES
S. Ramanujan, The Normal Number of Prime Factors of a Number n, reprinted at Chapter 35, Collected Papers (Hardy et al., ed), AMS Chelsea Publishing, 2000.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
EXAMPLE
R(3) = (PrimePi(4^2)-PrimePi(3^2)) / (PrimePi(5^2)-PrimePi(3^2)) = (PrimePi(16)-PrimePi(9)) / (PrimePi(25)-PrimePi(9)) = (6-4)/(9-4) = 2/5. Hence a(3) = 2. - Klaus Brockhaus, Jun 15 2009
MATHEMATICA
Numerator[(#[[2]]-#[[1]])/(#[[3]]-#[[1]])&/@Partition[PrimePi[ Range[ 90]^2], 3, 1]] (* Harvey P. Dale, Jan 06 2017 *)
PROG
(Magma) [ Numerator((#PrimesUpTo((n+1)^2)-a) / (#PrimesUpTo((n+2)^2)-a)) where a is #PrimesUpTo(n^2): n in [1..85] ]; // Klaus Brockhaus, Jun 15 2009
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Daniel Tisdale, Jun 14 2009
EXTENSIONS
a(1) inserted and extended beyond a(13) by Klaus Brockhaus, Jun 15 2009
Simplified title by John W. Nicholson, Dec 13 2013
STATUS
approved