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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 (list; graph; refs; listen; history; text; internal format)
 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. SolĂ©, 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. 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 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)  # 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) CROSSREFS Cf. A196668-A196675. Sequence in context: A093724 A130705 A051046 * A196673 A159027 A039492 Adjacent sequences:  A196664 A196665 A196666 * A196668 A196669 A196670 KEYWORD nonn AUTHOR Michel Planat, Oct 05 2011 STATUS approved

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