login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A165959 Size of the range of the Ramanujan Prime Corollary, 2*A168421(n) - A104272(n) 6
2, 3, 5, 5, 5, 11, 3, 7, 3, 9, 5, 11, 7, 9, 7, 11, 15, 13, 27, 25, 21, 15, 13, 11, 5, 17, 7, 3, 11, 9, 15, 9, 21, 13, 3, 15, 13, 7, 5, 15, 11, 11, 17, 15, 27, 21, 15, 13, 7, 21, 19, 15, 9, 3, 17, 15, 7, 7, 7, 9, 9, 17, 15, 11, 9, 5, 5, 21, 17, 11, 7, 15, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

All but the first term is odd because A104272 has only one even term, 2. Because of all primes > 2 are odd, 1 can be subtracted from each term.

If this sequence has an infinite number of terms in which a(n) = 3, then the twin prime conjecture can be proved.

R_n is the sequence A104272(n) and k = pi(R_n)= A000720(R_n) with i>k.

By comparing the fractions we can see that (p_(i+1)-p_i)/(2*Sqrt(p_i)) and a(n)/(2*Sqrt(p_k)) are < 1 for all n > 0, in fact a(n)/(1.8*Sqrt(p_k))< 1 for all n > 0. When taking into account numbers in A182873(n) and A190874(n) to the Sqrt(R_n) we see that A182873(n)/(A190874(n)*Sqrt(R_n)) < 1 for all n > 1.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009) 630-635.

J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2

Wikipedia, Ramanujan Prime

Marek Wolf, A Note on the Andrica Conjecture

FORMULA

a(n) = 2*A168421(n) - A104272(n)

EXAMPLE

A168421(19) = 127, A104272(19) = 227; so a(19) = 2*A168421(19) - A104272(19) = 254 - 227 = 27. Note: for n = 20, 21, 22, 23 A168421(n) = 127. Because A168421 remains the same for these n and A104272 increases, the size of the range for a(n) for these n decreases. Note: a(18) = 2*97 - 181 = 194 - 181 = 13. This is nearly half a(19). The actual gap betweens A104272(19) and the next prime, 229, is 2.

CROSSREFS

Cf. A168421, A104272, A182873, A190874.

Sequence in context: A003660 A065688 A169787 * A111164 A255347 A029910

Adjacent sequences:  A165956 A165957 A165958 * A165960 A165961 A165962

KEYWORD

nonn

AUTHOR

John W. Nicholson, Sep 12 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 4 21:33 EST 2016. Contains 278755 sequences.