%I #30 Sep 08 2022 08:45:59

%S 5,7,17,19,31,37,41,43,53,59,67,73,79,83,101,103,107,127,131,149,157,

%T 163,179,181,197,199,211,223,227,257,269,277,281,317,331,337,347,353,

%U 379,389,419,421,439,461,463,467,479,491,499,509,541,563,569,577,617

%N The Chebyshev primes of index 4.

%C The sequence consists of such odd prime numbers p that satisfy li(psi(p^4)) - li(psi(p^4-1)) < 1/4, where li(x) is the logarithmic integral and psi(x) is the Chebyshev psi function.

%H Dana Jacobsen, <a href="/A196670/b196670.txt">Table of n, a(n) for n = 1..75</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 A196670 := n -> PlanatSole(n,4); # _Peter Luschny_, Oct 23 2011

%t ChebyshevPsi[n_] := Log[LCM @@ Range[n]];

%t Reap[Do[If[LogIntegral[ChebyshevPsi[p^4]] - LogIntegral[ChebyshevPsi[p^4 - 1]] < 1/4, Print[p]; Sow[p]], {p, Prime[Range[2, 120]]}]][[2, 1]] (* _Jean-François Alcover_, Jul 14 2018, updated Dec 06 2018 *)

%o (Magma)

%o 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 jump4:=function(n);

%o x:=LogIntegral(tcheb(NthPrime(n)^4))-LogIntegral(tcheb(NthPrime(n)^4-1));

%o return x;

%o end function;

%o Set4:=[];

%o for i in [2..1000] do

%o if jump4(i)-1/4 lt 0 then Set4:=Append(Set4,NthPrime(i)); NthPrime(i); end if;

%o end for;

%o Set4;

%o (Sage)

%o def A196670(n) : return PlanatSole(n,4)

%o # The function PlanatSole(n,r) is in A196667.

%o # _Peter Luschny_, Oct 23 2011

%o (Perl) use ntheory ":all"; forprimes { say if 4 *(LogarithmicIntegral(chebyshev_psi($_**4)) - LogarithmicIntegral(chebyshev_psi($_**4-1))) < 1 } 3,100; # _Dana Jacobsen_, Dec 29 2015

%Y Cf. A196667, A196668, A196669.

%K nonn

%O 1,1

%A _Michel Planat_, Oct 05 2011

%E More terms from _Dana Jacobsen_, Dec 29 2015