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A196667
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The Chebyshev primes of index 1.
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10
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=1..52.
M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT]
L. Schoenfeld, Sharper bounds for the Chebyshev functions theta(x) and psi(x). II, Math. Comp. 30 (1975) 337-360.
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MAPLE
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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
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PROG
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(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) # Peter Luschny, Oct 23 2011
from mpmath import *
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 A196667(n) : return PlanatSole(n, 1)
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CROSSREFS
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Cf. A196668-A196675.
Sequence in context: A093724 A130705 A051046 * A196673 A159027 A039492
Adjacent sequences: A196664 A196665 A196666 * A196668 A196669 A196670
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KEYWORD
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nonn
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AUTHOR
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Michel Planat, Oct 05 2011
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STATUS
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approved
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