cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A196668 The Chebyshev primes of index 2.

Original entry on oeis.org

17, 29, 41, 53, 61, 71, 83, 89, 101, 103, 113, 127, 137, 149, 151, 157, 193, 211, 239, 241, 251, 257, 269, 293, 307, 311, 313, 317, 331, 353, 359, 373, 379, 389, 397, 401, 433, 439, 443, 457, 461, 463, 479, 499, 503, 509, 521, 523, 569, 571, 577, 587, 593, 599
Offset: 1

Views

Author

Michel Planat, Oct 05 2011

Keywords

Comments

The sequence consists of such odd prime numbers p that satisfy li[psi(p^2)]-li[psi(p^2-1)]<1/2, where li(x) is the logarithmic integral and psi(x) is the Chebyshev's psi function.

Links

Crossrefs

Cf. A196667, A196669, A196670.

Programs

  • 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;
jump2:=function(n);
x:=LogIntegral(tcheb(NthPrime(n)^2))-LogIntegral(tcheb(NthPrime(n)^2-1));
return x;
end function;
Set2:=[];
for i in [2..1000] do
if jump2(i)-1/2 lt 0 then Set2:=Append(Set2,NthPrime(i)); NthPrime(i); end if;
end for;
Set2;
  • Maple
    # The function PlanatSole(n,r) is in A196667.
A196668 := n -> PlanatSole(n,2); # Peter Luschny, Oct 23 2011
  • Mathematica
    ChebyshevPsi[n_] := Log[LCM @@ Range[n]];
Reap[Do[If[LogIntegral[ChebyshevPsi[p^2]] - LogIntegral[ChebyshevPsi[p^2 - 1]] < 1/2, Print[p]; Sow[p]], {p, Prime[Range[2, 200]]}]][[2, 1]] (* Jean-François Alcover, Jul 14 2018, updated Dec 06 2018 *)
  • Perl
    use ntheory ":all"; forprimes { say if 2 * (LogarithmicIntegral(chebyshev_psi($**2)) - LogarithmicIntegral(chebyshev_psi($**2-1))) < 1 } 3, 1000; # Dana Jacobsen, Dec 29 2015
  • Sage
    def A196668(n) : return PlanatSole(n,2)
# The function PlanatSole(n,r) is in A196667.
# Peter Luschny, Oct 23 2011

A196669 The Chebyshev primes of index 3.

Original entry on oeis.org

11, 19, 29, 61, 71, 97, 101, 107, 109, 113, 127, 131, 149, 151, 173, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 257, 269, 281, 307, 311, 313, 317, 347, 349, 359, 373, 383, 389, 401, 409, 419, 421, 433, 439, 461, 479, 503, 509, 557, 563, 569, 571, 607
Offset: 1

Views

Author

Michel Planat, Oct 05 2011

Keywords

Comments

The sequence consists of such odd prime numbers p that satisfy li[psi(p^3)]-li[psi(p^3-1)]<1/3, where li(x) is the logarithmic integral and psi(x) is the Chebyshev's psi function.

Links

Crossrefs

Cf. A196667, A196668, A196670.

Programs

  • 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;
jump3:=function(n);
x:=LogIntegral(tcheb(NthPrime(n)^3))-LogIntegral(tcheb(NthPrime(n)^3-1));
return x;
end function;
Set3:=[];
for i in [2..1000] do
if jump3(i)-1/3 lt 0 then Set3:=Append(Set3,NthPrime(i)); NthPrime(i); end if;
end for;
Set3;
  • Maple
    # The function PlanatSole(n,r) is in A196667.
A196669 := n -> PlanatSole(n,3); # Peter Luschny, Oct 23 2011
  • Mathematica
    ChebyshevPsi[n_] := Log[LCM @@ Range[n]];
Reap[Do[If[LogIntegral[ChebyshevPsi[p^3]] - LogIntegral[ChebyshevPsi[p^3 - 1]] < 1/3, Print[p]; Sow[p]], {p, Prime[Range[2, 120]]}]][[2, 1]] (* Jean-François Alcover, Jul 14 2018, updated Dec 06 2018 *)
  • Perl
    use ntheory ":all"; forprimes { say if 3 * (LogarithmicIntegral(chebyshev_psi($**3)) - LogarithmicIntegral(chebyshev_psi($**3-1))) < 1 } 3, 1000; # Dana Jacobsen, Dec 29 2015
  • Sage
    def A196669(n) : return PlanatSole(n,3)
# The function PlanatSole(n,r) is in A196667.
# Peter Luschny, Oct 23 2011

Extensions

Corrected and extended by Dana Jacobsen, Dec 29 2015

A197188 The Riemann primes of the psi type and index 4.

Original entry on oeis.org

2, 5, 7, 11, 13, 17, 31, 41, 43, 71, 89, 103, 109, 139, 173, 191, 197, 241, 281, 317, 443, 487, 569, 577, 701, 761, 797, 919, 1009
Offset: 1

Views

Author

Michel Planat, Oct 11 2011

Keywords

References

  • The sequence consists of the prime numbers p that are champions (left to right maxima) of the function |psi(p^4)-p^4|, where psi(p) is the Chebyshev psi function.

Links

Crossrefs

Cf. A196670, A197185, A197186, A197187.

Programs

  • Perl
    use ntheory ":all"; my($max,$f)=(0); forprimes { $f=abs(chebyshev_psi($**4)-$**4); if ($f > $max) { say; $max=$f; } } 1000; # Dana Jacobsen, Dec 28 2015

Extensions

More terms from Dana Jacobsen, Dec 28 2015

A197300 The Riemann primes of the theta type and index 4.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 47, 59, 61, 67, 71, 89, 97, 109, 137, 139, 167, 173, 191, 223, 229, 239, 241, 269, 271, 293, 311, 331, 347, 367, 401, 431, 433, 457, 503, 509, 571, 577, 661, 709, 719, 733, 739, 797, 911, 919, 1009
Offset: 1

Views

Author

Michel Planat, Oct 13 2011

Keywords

Comments

The sequence consists of the prime numbers p that are champions (left to right maxima) of the function |theta(p^4)-p^4|, where theta(p) is the Chebyshev theta function, theta(x) = sum_{primes p <=x } log p.

Links

Crossrefs

Cf. A035158, A196670, A197188, A197297, A197298, A197299.

Programs

  • Perl
    use ntheory ":all"; my($max,$f)=(0); forprimes { $f=abs(chebyshev_theta($**4)-$**4); if ($f > $max) { say; $max=$f; } } 1000; # Dana Jacobsen, Dec 28 2015

Extensions

More terms from Dana Jacobsen, Dec 28 2015
Showing 1-4 of 4 results.

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