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A196670 The Chebyshev primes of index 4. 5
5, 7, 17, 19, 31, 37, 41, 43, 53, 59, 67, 73, 79, 83, 101, 103, 107, 127, 131, 149, 157, 163, 179, 181, 197, 199, 211, 223, 227, 257, 269, 277, 281, 317, 331, 337, 347, 353, 379, 389, 419, 421, 439, 461, 463, 467, 479, 491, 499, 509, 541, 563, 569, 577, 617 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The sequence consists of such odd prime numbers p that satisfy li(psi(p^4)) - li(psi(p^4-1)) < 1/4, where li(x) is the logarithmic integral and psi(x) is the Chebyshev psi function.
LINKS
M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT], 2011.
MAPLE
# The function PlanatSole(n, r) is in A196667.
A196670 := n -> PlanatSole(n, 4); # Peter Luschny, Oct 23 2011
MATHEMATICA
ChebyshevPsi[n_] := Log[LCM @@ Range[n]];
Reap[Do[If[LogIntegral[ChebyshevPsi[p^4]] - LogIntegral[ChebyshevPsi[p^4 - 1]] < 1/4, Print[p]; Sow[p]], {p, Prime[Range[2, 120]]}]][[2, 1]] (* Jean-François Alcover, Jul 14 2018, 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;
jump4:=function(n);
x:=LogIntegral(tcheb(NthPrime(n)^4))-LogIntegral(tcheb(NthPrime(n)^4-1));
return x;
end function;
Set4:=[];
for i in [2..1000] do
if jump4(i)-1/4 lt 0 then Set4:=Append(Set4, NthPrime(i)); NthPrime(i); end if;
end for;
Set4;
(Sage)
def A196670(n) : return PlanatSole(n, 4)
# The function PlanatSole(n, r) is in A196667.
# Peter Luschny, Oct 23 2011
(Perl) use ntheory ":all"; forprimes { say if 4 *(LogarithmicIntegral(chebyshev_psi($_**4)) - LogarithmicIntegral(chebyshev_psi($_**4-1))) < 1 } 3, 100; # Dana Jacobsen, Dec 29 2015
CROSSREFS
Sequence in context: A106120 A106121 A185022 * A075304 A168245 A242929
KEYWORD
nonn
AUTHOR
Michel Planat, Oct 05 2011
EXTENSIONS
More terms from Dana Jacobsen, Dec 29 2015
STATUS
approved

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Last modified July 13 15:08 EDT 2024. Contains 374284 sequences. (Running on oeis4.)