

A233739


R(n)  p(2n), where R(n) is the nth Ramanujan prime and p(n) is the nth prime.


6



1, 4, 4, 10, 12, 10, 16, 14, 10, 26, 22, 18, 26, 42, 38, 36, 40, 30, 64, 56, 52, 46, 42, 40, 40, 42, 56, 48, 76, 68, 74, 62, 84, 72, 70, 72, 60, 56, 64, 78, 70, 70, 126, 114, 124, 114, 108, 98, 86, 100, 86, 78, 76, 66
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OFFSET

1,2


COMMENTS

a(n) = A104272(n)  A000040(2n).
The sequence tends to decrease at runs of Ramanujan primes and at twin Ramanujan primes.
Is 4 the minimum value of a(n) for all n > 1? Is the sequence unbounded? What are its liminf and limsup? Is a(n)/n bounded?
Christian Axler has proved that the answers to the 1st, 2nd, and 4th questions are yes, and that liminf a(n) = limsup a(n) = infinity.  Jonathan Sondow, Feb 12 2014
a(n) > n, for 1 < n < 86853959 = limit. For limit, a(n) = 135595760, a(n)  n = 48741801.  John W. Nicholson, Dec 19 2013


LINKS

John W. Nicholson, Table of n, a(n) for n = 1..10000
Christian Axler, On generalized Ramanujan primes, arXiv:1401.7179 [math.NT], 2014.
J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009), 630635.
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


FORMULA

a(n) >= 2 for n > 1 (see "Ramanujan primes and Bertrand's postulate").
a(n)/p(2n) = R(n)/p(2n)  1 > 0 as n > infinity (see same link).


EXAMPLE

R(2)  p(4) = 11  7 and R(3)  p(6) = 17  13, so a(2) = a(3) = 4.


MATHEMATICA

nn = 60; R = Table[0, {nn}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s]; If[s < nn, R[[s + 1]] = k], {k, Prime[3 nn]}];
R = R + 1;
Table[R[[n]]  Prime[2 n], {n, 1, nn}] (* JeanFrançois Alcover, Nov 07 2018, using T. D. Noe's code for R *)


CROSSREFS

a(n) = 2*A233740(n) for n > 1. Records are A233741.
Sequence in context: A219939 A219471 A006477 * A279036 A182699 A058596
Adjacent sequences: A233736 A233737 A233738 * A233740 A233741 A233742


KEYWORD

sign


AUTHOR

Jonathan Sondow, Dec 15 2013


STATUS

approved



