A161621 Numerator of (b(n+1) - b(n))/(b(n+2) - b(n)), where b(n) = A038107(n) is the number of primes up to n^2.

1, 1, 2, 3, 1, 4, 3, 4, 3, 5, 4, 1, 5, 2, 6, 7, 5, 1, 6, 1, 1, 7, 2, 9, 8, 7, 8, 9, 1, 4, 10, 9, 10, 9, 10, 1, 3, 12, 11, 12, 11, 3, 12, 11, 13, 10, 13, 3, 10, 11, 15, 4, 12, 13, 11, 12, 17, 13, 1, 16, 13, 17, 15, 7, 16, 1, 15, 17, 13, 7, 1, 15, 1, 17, 9, 11, 7, 18, 23, 13, 20, 19, 20, 17, 16

Offset: 1

Author

Daniel Tisdale, Jun 14 2009

Keywords

• nonn
• frac

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, ...}.

Examples

			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
		

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

Crossrefs

Cf. A014085

Cf. A161622 (denominators). - Klaus Brockhaus, Jun 15 2009

Programs

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

Mathematica

Numerator[(#[[2]]-#[[1]])/(#[[3]]-#[[1]])&/@Partition[PrimePi[ Range[ 90]^2],3,1]] (* Harvey P. Dale, Jan 06 2017 *)

Extensions

a(1) inserted and extended beyond a(13) by Klaus Brockhaus, Jun 15 2009

Simplified title by John W. Nicholson, Dec 13 2013