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%I #34 Sep 08 2022 08:45:59
%S 11,19,29,61,71,97,101,107,109,113,127,131,149,151,173,181,191,193,
%T 197,199,211,223,227,229,233,257,269,281,307,311,313,317,347,349,359,
%U 373,383,389,401,409,419,421,433,439,461,479,503,509,557,563,569,571,607
%N The Chebyshev primes of index 3.
%C The sequence consists of such odd prime numbers p that satisfy li[psi(p^3)]-li[psi(p^3-1)]<1/3, where li(x) is the logarithmic integral and psi(x) is the Chebyshev's psi function.
%H Dana Jacobsen, <a href="/A196669/b196669.txt">Table of n, a(n) for n = 1..314</a>
%H 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 # The function PlanatSole(n,r) is in A196667.
%p A196669 := n -> PlanatSole(n,3); # _Peter Luschny_, Oct 23 2011
%t ChebyshevPsi[n_] := Log[LCM @@ Range[n]];
%t Reap[Do[If[LogIntegral[ChebyshevPsi[p^3]] - LogIntegral[ChebyshevPsi[p^3 - 1]] < 1/3, Print[p]; Sow[p]], {p, Prime[Range[2, 120]]}]][[2, 1]] (* _Jean-François Alcover_, Jul 14 2018, updated Dec 06 2018 *)
%o (Magma) Mangoldt:=function(n);
%o if #Factorization(n) eq 1 then return Log(Factorization(n)[1][1]); else return 0; end if;
%o end function;
%o tcheb:=function(n);
%o x:=0;
%o for i in [1..n] do
%o x:=x+Mangoldt(i);
%o end for;
%o return(x);
%o end function;
%o jump3:=function(n);
%o x:=LogIntegral(tcheb(NthPrime(n)^3))-LogIntegral(tcheb(NthPrime(n)^3-1));
%o return x;
%o end function;
%o Set3:=[];
%o for i in [2..1000] do
%o if jump3(i)-1/3 lt 0 then Set3:=Append(Set3,NthPrime(i)); NthPrime(i); end if;
%o end for;
%o Set3;
%o (Sage)
%o def A196669(n) : return PlanatSole(n,3)
%o # The function PlanatSole(n,r) is in A196667.
%o # _Peter Luschny_, Oct 23 2011
%o (Perl) use ntheory ":all"; forprimes { say if 3 * (LogarithmicIntegral(chebyshev_psi($_**3)) - LogarithmicIntegral(chebyshev_psi($_**3-1))) < 1 } 3, 1000; # _Dana Jacobsen_, Dec 29 2015
%Y Cf. A196667, A196668, A196670.
%K nonn
%O 1,1
%A _Michel Planat_, Oct 05 2011
%E Corrected and extended by _Dana Jacobsen_, Dec 29 2015
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