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 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 pp_n, contain the same number of primes, and a(n)=0, if no such prime p exists. 5
 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 (list; graph; refs; listen; history; text; internal format)
 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 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, 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]; For[p = 2, True, p = NextPrime[p], Which[ppn, 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: A057435 A246728 A155685 * 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

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)