A196675 -id:A196675 - cp's OEIS frontend

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

Michel Planat, Oct 05 2011

nonn

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.

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.

Cf. A196668-A196675.

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;

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

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 *)

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

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

A196674 Numbers n such that c(n) = p_{2n}, where c(n) is the n-th Chebyshev prime and p_{2n} the 2n-th prime.

Original entry on oeis.org

510, 10271, 11259, 11987, 14730, 18772, 18884, 21845, 24083, 33723, 46789

Offset: 1

Michel Planat, Oct 05 2011

nonn

The Chebyshev primes (of index 1) are such odd primes that satisfy li[psi(p)]-li[psi(p-1)]<1 (sequence A196667), where li(x) is the logarithmic integral and psi(x) is the Chebyshev's psi function.
THe present sequence lists the zeros of the function c(n)-p_{2n}, where c(n) is the n-th Chebyshev prime and p_{2n} the 2n-th prime.
See A196675 for the Chebyshev primes satisfying a(n)=p_{2n}.

M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT]

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A196667 The Chebyshev primes of index 1.

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