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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 - 1 = 26. 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 - 1 = 194 - 181 - 1 = 12. This is less than 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: A246401 A003660 A169787 * A111164 A029910 A063677

Adjacent sequences:  A165956 A165957 A165958 * A165960 A165961 A165962

KEYWORD

nonn

AUTHOR

John W. Nicholson, Sep 12 2011

STATUS

approved

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Last modified October 25 23:29 EDT 2014. Contains 248566 sequences.