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A196668
The Chebyshev primes of index 2.
6
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
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
KEYWORD
nonn
AUTHOR
Michel Planat, Oct 05 2011
STATUS
approved