|
Search: id:A161621
|
|
|
| A161621 |
|
Numerators of the ratios (in lowest terms) of numbers of primes in one square interval to that of the interval and its successor. The numerators are derived from sequence A014085. The expression is: R(n)=(PrimePi[(n+1)^2] - PrimePi[n^2])/(PrimePi[(n+2)^2] - PrimePi[n^2]); The first few ratios are: 1/2, 2/5, 3/5, 1/3, 4/7,... |
|
+0
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; listen)
|
|
|
OFFSET
|
1,3
|
|
|
COMMENT
|
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 ref. [1], below. The PNT and ref. [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
|
|
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. [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 15 2009]
|
|
PROGRAM
|
(MAGMA) [ Numerator((#PrimesUpTo((n+1)^2)-a) / (#PrimesUpTo((n+2)^2)-a)) where a is #PrimesUpTo(n^2): n in [1..85] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 15 2009]
|
|
CROSSREFS
|
Cf. A014085
Cf. A161622 (denominators). [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 15 2009]
Sequence in context: A006021 A002186 A125936 this_sequence A095701 A067992 A140757
Adjacent sequences: A161618 A161619 A161620 this_sequence A161622 A161623 A161624
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Daniel Tisdale (daniel6874(AT)gmail.com), Jun 14 2009
|
|
EXTENSIONS
|
a(1) inserted and extended beyond a(13) by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 15 2009
|
|
|
Search completed in 0.002 seconds
|
