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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. 7
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 (list; graph; refs; listen; history; text; internal format)
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

Cf. A014085

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

Sequence in context: A243614 A200942 A286234 * A095701 A067992 A317024

Adjacent sequences:  A161618 A161619 A161620 * A161622 A161623 A161624

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

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Last modified October 15 01:40 EDT 2019. Contains 328025 sequences. (Running on oeis4.)