A164554 Ramanujan primes A104272(n) for which A104272(n) = A080359(n).

2, 71, 101, 181, 239, 241, 269, 349, 373, 409, 419, 433, 439, 491, 593, 599, 601, 607, 647, 653, 659, 823, 827, 857, 947, 1021, 1031, 1061, 1063, 1091, 1103, 1301, 1427, 1429, 1447, 1451, 1489, 1553, 1559, 1567, 1601, 1607, 1609, 1789, 1867, 1871, 1913, 1999, 2003

Offset

1,1

Comments

For every n>=1, A104272(n) >= A080359(n), and the sequence shows where the inequality becomes an equality.

Let prime(m) < a(n)/2 < prime(m+1); then there exist primes p<q such that p is in the interval (2*Prime(m), a(n)) and q is in the interval (a(n), 2*Prime(m+1)).

For example, a(2) = 71, 31 < a(2)/2 < 37 and intervals (62,71), (71,74) contain the primes p = 67 and q = 73 respectively.

Let us call a prime p compatible with another prime q, if the intervals (p/2,q/2) and (p,q], if q>p, (or intervals (q/2,p/2) and (q,p], if q<p) contain the same number of primes. If p is compatible with no other prime, we call it a peculiar prime. The sequence lists the peculiar primes. [Vladimir Shevelev, Apr 25 2012]

Links

Table of n, a(n) for n=1..49.

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT] [From Vladimir Shevelev, Aug 20 2009]

V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4

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

All solutions of the equation A104272(x)=A080359(x) are x=pi(a(n))-pi(a(n)/2). - Vladimir Shevelev, Apr 25 2012

Example

a(2)=71, such that 31<71/2<37, and we see that p=67 is in interval (62, 71) and q=73 is in interval (71, 74).

Mathematica

nn = 200; t = Table[0, {nn+1}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s <= nn && t[[s+1]] == 0, t[[s+1]] = k], {k, Prime[3nn]}
];
A080359 = Rest[t];
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[3nn]}
];
A104272 = R+1;
Intersection[A104272, A080359] (* Jean-François Alcover, Oct 28 2018, after T. D. Noe in A104272 *)

Crossrefs

Cf. A104272, A080359, A164368, A164333, A164288, A164294, A212493, A212541.

Sequence in context: A348456 A358275 A293753 * A141908 A234298 A157368

Adjacent sequences: A164551 A164552 A164553 * A164555 A164556 A164557

Keyword

nonn

Author

Vladimir Shevelev, Aug 15 2009

Extensions

Terms beyond 659 from R. J. Mathar, Dec 17 2009

Status

approved