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A196668 The Chebyshev primes of index 2.

%I A196668 #33 Sep 08 2022 08:45:59
%S A196668 17,29,41,53,61,71,83,89,101,103,113,127,137,149,151,157,193,211,239,
%T A196668 241,251,257,269,293,307,311,313,317,331,353,359,373,379,389,397,401,
%U A196668 433,439,443,457,461,463,479,499,503,509,521,523,569,571,577,587,593,599
%N A196668 The Chebyshev primes of index 2.
%C A196668 The sequence consists of such odd prime numbers p that satisfy li[psi(p^2)]-li[psi(p^2-1)]<1/2, where li(x) is the logarithmic integral and psi(x) is the Chebyshev's psi function.
%H A196668 Dana Jacobsen, <a href="/A196668/b196668.txt">Table of n, a(n) for n = 1..4035</a>
%H A196668 M. Planat and P. Solé, <a href="http://arxiv.org/abs/1109.6489">Efficient prime counting and the Chebyshev primes</a> arXiv:1109.6489 [math.NT], 2011.
%p A196668 # The function PlanatSole(n,r) is in A196667.
%p A196668 A196668 := n -> PlanatSole(n,2); # _Peter Luschny_, Oct 23 2011
%t A196668 ChebyshevPsi[n_] := Log[LCM @@ Range[n]];
%t A196668 Reap[Do[If[LogIntegral[ChebyshevPsi[p^2]] - LogIntegral[ChebyshevPsi[p^2 - 1]] < 1/2, Print[p]; Sow[p]], {p, Prime[Range[2, 200]]}]][[2, 1]] (* _Jean-François Alcover_, Jul 14 2018, updated Dec 06 2018 *)
%o A196668 (Magma) Mangoldt:=function(n);
%o A196668 if #Factorization(n) eq 1 then return Log(Factorization(n)[1][1]); else return 0; end if;
%o A196668 end function;
%o A196668 tcheb:=function(n);
%o A196668 x:=0;
%o A196668 for i in [1..n] do
%o A196668 x:=x+Mangoldt(i);
%o A196668 end for;
%o A196668 return(x);
%o A196668 end function;
%o A196668 jump2:=function(n);
%o A196668 x:=LogIntegral(tcheb(NthPrime(n)^2))-LogIntegral(tcheb(NthPrime(n)^2-1));
%o A196668 return x;
%o A196668 end function;
%o A196668 Set2:=[];
%o A196668 for i in [2..1000] do
%o A196668 if jump2(i)-1/2 lt 0 then Set2:=Append(Set2,NthPrime(i)); NthPrime(i); end if;
%o A196668 end for;
%o A196668 Set2;
%o A196668 (Sage)
%o A196668 def A196668(n) : return PlanatSole(n,2)
%o A196668 # The function PlanatSole(n,r) is in A196667.
%o A196668 # _Peter Luschny_, Oct 23 2011
%o A196668 (Perl) use ntheory ":all"; forprimes { say if 2 * (LogarithmicIntegral(chebyshev_psi($_**2)) - LogarithmicIntegral(chebyshev_psi($_**2-1))) < 1 } 3, 1000; # _Dana Jacobsen_, Dec 29 2015
%Y A196668 Cf. A196667, A196669, A196670.
%K A196668 nonn
%O A196668 1,1
%A A196668 _Michel Planat_, Oct 05 2011