A212493 Let p_n=prime(n), n>=1. Then a(n) is the least prime p which differs from p_n, for which the intervals (p/2,p_n/2), (p,p_n], if p<p_n, or the intervals (p_n/2,p/2), (p_n,p], if p>p_n, contain the same number of primes, and a(n)=0, if no such prime p exists.

0, 5, 3, 3, 3, 17, 13, 23, 19, 19, 37, 31, 31, 47, 43, 59, 53, 67, 61, 0, 79, 73, 73, 73, 73, 0, 107, 103, 127, 131, 109, 113, 113, 151, 113, 139, 163, 157, 157, 179, 173, 0, 223, 197, 193, 233, 193, 191, 191, 193, 199, 0, 0, 257, 251, 251, 0, 277, 271, 271

Offset

1,2

Comments

a(n)=0 if and only if p_n is a peculiar prime, i.e., simultaneously Ramanujan (A104272) and Labos (A080359) prime (see sequence A164554).

a(n)>p_n if and only if p_n is Labos prime but not Ramanujan prime.

Links

Table of n, a(n) for n=1..60.

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

Jonathan Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.

Formula

If p_n is not a Labos prime, then a(n) = A080359(n-pi(p_n/2)).

Example

Let n=5, p_5=11; p=2 is not suitable, since in (1,5.5) we have 3 primes, while in (2,11] there are 4 primes. Consider p=3. Now in intervals (1.5,5.5) and (3,11] we have the same number (3) of primes. Therefore, a(5)=3. The same value we can obtain by the formula. Since 11 is not a Labos prime, then a(5)=A080359(5-pi(5.5))=A080359(2)=3.

Mathematica

terms = 60; nn = Prime[terms];
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 + 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[3 nn]}];
A080359 = Rest[t];
a[n_] := Module[{}, pn = Prime[n]; If[MemberQ[A104272, pn] && MemberQ[ A080359, pn], Return[0]]; For[p = 2, True, p = NextPrime[p], Which[p<pn, If[PrimePi[pn/2] - PrimePi[p/2] == PrimePi[pn] - PrimePi[p], Return[p]], p>pn, If[PrimePi[p/2] - PrimePi[pn/2] == PrimePi[p] - PrimePi[pn], Return[p]]]]];
Array[a, terms] (* Jean-François Alcover, Dec 04 2018, after T. D. Noe in A104272 *)

Crossrefs

Cf. A104272, A080359, A164554.

Sequence in context: A246728 A155685 A359199 * A011320 A208123 A090489

Adjacent sequences: A212490 A212491 A212492 * A212494 A212495 A212496

Keyword

nonn

Author

Vladimir Shevelev and Peter J. C. Moses, May 18 2012

Status

approved