A196667 The Chebyshev primes of index 1.

109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, 467, 479, 491, 503, 509, 523, 619, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 761, 769, 773, 829, 859, 863, 883, 887, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1091, 1093, 1097

Offset: 1

Author

Michel Planat, Oct 05 2011

Keywords

• nonn

Comments

The sequence consists of the odd prime numbers p that satisfy li[psi(p)]-li[psi(p-1)]<1, 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..10000
• M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT], 2011.
• L. Schoenfeld, Sharper bounds for the Chebyshev functions theta(x) and psi(x). II, Math. Comp. 30 (1975) 337-360.

Crossrefs

Cf. A196668-A196675.

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;
jump1:=function(n);
x:=LogIntegral(tcheb(NthPrime(n)))-LogIntegral(tcheb(NthPrime(n)-1));
return x;
end function;
Set1:=[];
for i in [2..1000] do
if jump1(i)-1 lt 0 then Set1:=Append(Set1,NthPrime(i)); NthPrime(i); end if;
end for;
Set1;

Maple

PlanatSole := proc(n,r) local j, p, pr, psi, L; L := NULL;
psi := n -> add(log(i/ilcm(op(numtheory[divisors](i) minus {1,i}))),i=1..n);
for j in [$3..n] do p := ithprime(j); pr := p^r;
if evalf(Li(psi(pr))-Li(psi(pr-1))) < 1/r then L:= L,p fi od; L end:
A196667 := n -> PlanatSole(n,1); # Peter Luschny, Oct 23 2011

Mathematica

ChebyshevPsi[n_] := Log[LCM @@ Range[n]];
Reap[Do[If[LogIntegral[ChebyshevPsi[p]] - LogIntegral[ChebyshevPsi[p - 1]] < 1, Sow[p]], {p, Prime[Range[2, 200]]}]][[2, 1]] (* Jean-François Alcover, Nov 17 2017, updated Dec 06 2018 *)

Perl

use ntheory ":all"; forprimes { say if LogarithmicIntegral(chebyshev_psi($))-LogarithmicIntegral(chebyshev_psi($-1)) < 1 } 3,1000; # Dana Jacobsen, Dec 29 2015

Sage

from mpmath import mp, mangoldt
mp.dps = 25;
def psi(n) :
    return sum(mangoldt(i) for i in (1..n))
def PlanatSole(n,r) :
    P = Primes(); L = []
    for j in (2..n):
        p = P.unrank(j)
        pr = p^r
        if Li(psi(pr)) - Li(psi(pr-1)) < 1/r :
           L.append(p)
    return L
def A196667List(n) : return PlanatSole(n,1)
A196667List(100) # Peter Luschny, Oct 23 2011