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

1,1

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.

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 *)

Prog

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

Crossrefs

Cf. A196667, A196669, A196670.

Sequence in context: A087937 A103739 A255871 * A096785 A104228 A049484

Adjacent sequences: A196665 A196666 A196667 * A196669 A196670 A196671

Keyword

nonn

Author

Michel Planat, Oct 05 2011

Status

approved