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A196667
The Chebyshev primes of index 1.
10
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
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
KEYWORD
nonn
AUTHOR
Michel Planat, Oct 05 2011
STATUS
approved