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%I #35 Apr 08 2024 06:55:48
%S 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,
%T 0,107,103,127,131,109,113,113,151,113,139,163,157,157,179,173,0,223,
%U 197,193,233,193,191,191,193,199,0,0,257,251,251,0,277,271,271
%N 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.
%C 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).
%C a(n)>p_n if and only if p_n is Labos prime but not Ramanujan prime.
%H Vladimir Shevelev, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos primes, their generalizations, and classifications of primes</a>, J. Integer Seq. 15 (2012) Article 12.5.4.
%H Jonathan Sondow, J. W. Nicholson, and T. D. Noe, <a href="http://arxiv.org/abs/1105.2249"> Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
%F If p_n is not a Labos prime, then a(n) = A080359(n-pi(p_n/2)).
%e 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.
%t terms = 60; nn = Prime[terms];
%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[3 nn]}];
%t A104272 = R + 1;
%t 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]}];
%t A080359 = Rest[t];
%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]]]]];
%t Array[a, terms] (* _Jean-François Alcover_, Dec 04 2018, after _T. D. Noe_ in A104272 *)
%Y Cf. A104272, A080359, A164554.
%K nonn
%O 1,2
%A _Vladimir Shevelev_ and _Peter J. C. Moses_, May 18 2012
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