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

1,1

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.

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 *)

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

Crossrefs

Cf. A196668-A196675.

Sequence in context: A263194 A231701 A051046 * A196673 A159027 A039492

Adjacent sequences: A196664 A196665 A196666 * A196668 A196669 A196670

Keyword

nonn

Author

Michel Planat, Oct 05 2011

Status

approved