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
KEYWORD
nonn
AUTHOR
Michel Planat, Oct 05 2011
STATUS
approved