#A52 INTEGERS 15 (2015) AN IMPROVED UPPER BOUND FOR RAMANUJAN PRIMES

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1 #A5 INTEGERS 15 (015) AN IMPROVED UPPER BOUND FOR RAMANUJAN PRIMES Anitha Srinivasan Dept. of Mathematics, Saint Louis University Madrid Campus, Madrid, Spain John W. Nicholson Decatur, Mississippi Received: 1//14, Revised: 10/8/15, Accepted: 1//15, Published: 1/11/15 Abstract For n 1, the n th Ramanujan prime is defined as the least positive inteer R n such that for all x R n, the interval ( x, x] has at least n primes. If = n 1 +, then we show that R n < p [ ] for all n > 41, where p i lo n + lo n 4 is the i th prime. This bound improves upon all previous bounds for lare n. 1. Introduction For n 1, the n th Ramanujan prime is defined as the least positive inteer R n, such that for all x R n, the interval ( x, x] has at least n primes. Note that by the minimality condition, R n is prime and the interval ( Rn, R n] contains exactly n primes. Let p n denote the n th prime. Sondow [] showed that p n < R n < p 4n for all n and conjectured that R n < p n for all n. This conjecture was proved by Laishram [] and subsequently Sondow, Nicholson and Noe [4] improved Laishram s result by showin that R n < p n. Axler [1, Proposition.4] showed that for t > we have R n apple p [tn] for all n, where [x] denotes the inteer part of x. In [5] it was shown that for every > 0, there exists an inteer N such that R n < p [n(1+ )] for all n > N. Our main result below ives a new upper bound that, for lare n, is better than all previous bounds. Theorem 1. Let R n = p s be the n th Ramanujan prime, where p s is the s th prime. Then s < n 1 + lo n+lo(lo n) 4 for all n > 41. Note that while all previous upper bounds for R n are of the form of p [nc], where c > 1 is a constant, the bound iven in the theorem above is of the form p [nf(n)], where f(n) > 1 and lim n!1 f(n) = 1. Hence for lare n our bound is smaller and thus better than p [nc] for any fixed c.

2 INTEGERS: 15 (015). Proof of Theorem 1 The proof follows closely the proof of the main theorem in [5], where the followin function was defined. Let F (x) = x(lo x + lo lo x) (x n)(lo(x n) + lo lo(x n) 1), where n is a fixed positive inteer. Henceforth we denote lo(lo n) by lo n. Also, let (x) denote the number of primes less than or equal to x. In [5] we showed that for a Ramanujan prime R n = p s we have p s n < ps. A similar result, with the inequality reversed, holds for indices reater than s. We include both of these results in the followin lemma, which while not used directly in the proofs here, is of independent interest and relevant to the current topic. The result in the second part of the lemma has been dubbed as the Ramanujan prime corollary by the second author (see the first three sequences and last sequence iven in Table 1 of the Appendix). Lemma 1. Let R n = p s be the n th Ramanujan prime, where p s is the s th prime. Then the followin hold: (i) p s n < ps for all n. [5, Lemma.1]. (ii) p s+k < p s+k n for all positive inteers k. Proof. Let i = s + k. Note that by definition of R n we have pi pi 1 (p i 1) = (p i 1) n. Therefore, p i 1 p i 1 > p i > p i >... > p i n > pi, and hence p i n > pi and the second part of the lemma follows notin that i = s + k. Lemma. Let R n = p s be the n th Ramanujan prime, where p s is the s th prime. Then the followin hold: (i) n < s < n ([, ]). (ii) F (s) > 0 and F (x) is a decreasin function for all x 1.1]. n [5, Proof of Theorem Lemma. Let = (n) = lo n + lo n 4. Then for all n > 41 we have F n (n) < 0. Proof. We first observe that for n > 41 we have > 1. Let = lo n lo n 1 + 1

3 INTEGERS: 15 (015) and = lo n lo n Observe that = 1 + lo + lo! lo 1 n + n A lo n + n < + lo <, 1+ 1 lo(n+ as lo < 0 and 0 < lo ) < 1. Therefore, 1+ lo(n+ n ) = ( ) > and hence > > lo n + lo n 4 =, which ives and the lemma follows as F < 1 +, n = n n 1 +. Proof of Theorem 1 Let R n = p s. Then by Lemma, part (i), we have s > n and F (s) > 0. Moreover, by Lemma, part (ii), F (x) is a decreasin function for x n. Let = (n) = lo n+lo n 4 and = n(1 + 1 ). Then by Lemma we have F ( ) < 0 for all n > 41. As F is a decreasin function for x n and F (s) > 0, we have s apple n(1 + 1 ) for all n > 41. References [1] C. Axler, On eneralized Ramanujan primes, Arxiv: v1. [] S. Laishram, On a conjecture on Ramanujan primes, Int. J. Number Theory 6, (010), [] J. Sondow, Ramanujan primes and Bertrand s postulate, Amer. Math. Monthly 116, (009), [4] J. Sondow, J. W. Nicholson, T. D. Noe, Ramanujan primes: bounds, runs, twins, and aps, J. Inteer Sequences 14, (011), Article [5] A. Srinivasan, An upper bound for Ramanujan primes, Inteers 14 (014).

4 INTEGERS: 15 (015) 4 Appendix: Tables of Sequences from the OEIS at Sequence Title A Size of the rane of the Ramanujan Prime Corollary, *A16841(n) - A1047(n). A16841 Small Associated Ramanujan Prime, p i n. A16845 Lare Associated Ramanujan Prime, p i. A17460 Smallest prime that beins a run of n Ramanujan primes that are consecutive primes. A17465 A Prime numbers that are not Ramanujan primes. Smallest prime that beins a run of n consecutive primes that are not Ramanujan primes. A Number of primes up to the n th Ramanujan prime: A19014 A00070(A1047(n)), (R n). Decimal expansion of Ramanujan prime constant: P 1 n=1 1/R n, where R n is the n th Ramanujan prime, A1047(n). A1900 Decimal expansion of sum of alternatin series of reciprocals of Ramanujan primes, P 1 n=1 (1/Rn)( 1)n 1, where R n is the n th Ramanujan prime, A1047(n). A Number of Ramanujan primes R k such that (n 1) < R k apple n. A19050 Number of Ramanujan primes apple n. A First di erences of A179196, (R (n+1) ) (R n) where R n is A1047(n). A1915 Number of Ramanujan primes R k between trianular numbers T (n 1) < R k apple T (n). A1916 First occurrence of number n of Ramanujan primes in A1915. A1917 Last known occurrence of number n of Ramanujan primes in A1915. A1918 Greatest Ramanujan prime index less than x. A14756 a(n) = larest Ramanujan prime R k in A1047 that is apple A0086(n). A14757 a(n) = smallest Ramanujan prime R k in A1047 that is A000101(n). A1496 Di erence A1495(n) - A1494(n), prime count between Ramanujan primes boundin maximal ap primes. A1494 Numbers R(k) such that R(k) k lo R(k), where R(k) = A1047(k) is the k th Ramanujan prime. A79 R(n) p(n), where R(n) is the n th Ramanujan prime and p(n) is the n th prime.. A498 Ramanujan prime R k such that (R k+1 ) (R k ) are record values: record Ramanujan prime A190874(k). Table 1: Sequences authored by John W. Nicholson

5 INTEGERS: 15 (015) 5 Sequence Title Comments A Increasin aps between primes (upper end) (compare with A0086, which ives lower ends of these aps). A0058 Primes p such that p 1 is also prime. Except for a(1)= and a()=5, a(n) = A16841(k). Primes and 5 are special in that they are the only primes which do not have a Ramanujan prime between them and their double, apple 6 and 10 respectively. Because of the lare size of a ap, there are many repeats of the prime number in A John W. Nicholson, Dec If a(n) is in A16841 then A0058(n) is a twin prime with a Ramanujan prime, A0058(n) -. If this sequence has an infinite number of terms in A16841, then the twin prime conjecture can be proved. - John W. Nicholson, Dec A1047 A Ramanujan primes R n: a(n) is the smallest number such that if x apple a(n), then (x) (x/) apple n, where (x) is the number of primes apple x. Least number a(n) such that there are at least n primes in the rane (T (k 1), T (k)] for all k a(n), where T(k) is the k-th trianular number. For some n and k, we see that A16841(k) = a(n) so as to form a chain of primes similar to a Cunninham chain. For example (and the first example), A16841() = 7, links a() = 11 = A16841(), links a() = 17 = A16841(4), links a(4) = 9 = A16841(6), links a(6) = 47. Note that the links do not have to be of a form like q = *p+1 or q = *p-1. - John W. Nicholson, Feb 015 With R n the n-th Ramanujan prime (A1047), it is conjectured that for every n 0, (1/)R n apple a(n) < (0/1)R n. These bounds have been verified for all n up to For most n apple 8000, we have a(n) > R n, with exceptions listed in A Table : Observations (four sequences)

6 INTEGERS: 15 (015) 6 Sequence Title A04814 Number of decompositions of n into an unordered sum of two Ramanujan primes. A05616 Even numbers that are not the sum of two non-ramanujan primes (A17465). COMMENTS: No other terms < Conjectured to be complete. A05617 Number of decompositions of n into an unordered sum of two non- Ramanujan primes (A17465). A05618 Last occurrence of n partitions in A Table : Authored by (or suested by) John W. Nicholson and Donovan Johnson (4 sequences). Note: All are related to Goldbach conjecture.