A165959 Size of the range of the Ramanujan Prime Corollary, 2*A168421(n) - A104272(n).

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

Offset: 1

Author

John W. Nicholson, Sep 12 2011

Keywords

• nonn

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 sqrt(R_n) we see that A182873(n)/(A190874(n)*sqrt(R_n)) < 1 for all n > 1.

Examples

			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.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-201; Amer. Math. Monthly 116 (2009) 630-635.
• Jonathan Sondow, John W. Nicholson, and Tony D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
• Wikipedia, Ramanujan Prime
• Marek Wolf, A Note on the Andrica Conjecture, arXiv:1010.3945 [math.NT], 2010.

Crossrefs

Cf. A168421, A104272, A182873, A190874.

Programs

Mathematica

nn = 100; 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]}];
A104272 = R + 1; t = Table[0, {nn}];
Do[m = PrimePi[2 n] - PrimePi[n]; If[0 < m <= nn, t[[m]] = n], {n, 15 nn}];
A168421 = NextPrime[Join[{1}, t]] // Most;
A165979 = 2 A168421 - A104272 (* Jean-François Alcover, Nov 07 2018, after T. D. Noe in A104272 *)

Formula

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