A196668 The Chebyshev primes of index 2.

17, 29, 41, 53, 61, 71, 83, 89, 101, 103, 113, 127, 137, 149, 151, 157, 193, 211, 239, 241, 251, 257, 269, 293, 307, 311, 313, 317, 331, 353, 359, 373, 379, 389, 397, 401, 433, 439, 443, 457, 461, 463, 479, 499, 503, 509, 521, 523, 569, 571, 577, 587, 593, 599

Offset: 1

Author

Michel Planat, Oct 05 2011

Keywords

• nonn

Comments

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.

Links

• Dana Jacobsen, Table of n, a(n) for n = 1..4035
• M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT], 2011.

Crossrefs

Cf. A196667, A196669, A196670.

Programs

Magma

Mangoldt:=function(n);
if #Factorization(n) eq 1 then return Log(Factorization(n)[1][1]); else return 0; end if;
end function;
tcheb:=function(n);
x:=0;
for i in [1..n] do
x:=x+Mangoldt(i);
end for;
return(x);
end function;
jump2:=function(n);
x:=LogIntegral(tcheb(NthPrime(n)^2))-LogIntegral(tcheb(NthPrime(n)^2-1));
return x;
end function;
Set2:=[];
for i in [2..1000] do
if jump2(i)-1/2 lt 0 then Set2:=Append(Set2,NthPrime(i)); NthPrime(i); end if;
end for;
Set2;

Maple

# The function PlanatSole(n,r) is in A196667.
A196668 := n -> PlanatSole(n,2); # Peter Luschny, Oct 23 2011

Mathematica

ChebyshevPsi[n_] := Log[LCM @@ Range[n]];
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 *)

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

Sage

def A196668(n) : return PlanatSole(n,2)
# The function PlanatSole(n,r) is in A196667.
# Peter Luschny, Oct 23 2011