A196668 -id:A196668 - cp's OEIS frontend

A196667 The Chebyshev primes of index 1.

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

Offset: 1

Michel Planat, Oct 05 2011

nonn

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.

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.

Cf. A196668-A196675.

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;

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

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 *)

use ntheory ":all"; forprimes { say if LogarithmicIntegral(chebyshev_psi($))-LogarithmicIntegral(chebyshev_psi($-1)) < 1 } 3,1000; # Dana Jacobsen, Dec 29 2015

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

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

Michel Planat, Oct 05 2011

nonn

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.

Dana Jacobsen, Table of n, a(n) for n = 1..314
M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT], 2011.

Cf. A196667, A196668, A196670.

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;

# The function PlanatSole(n,r) is in A196667.
A196669 := n -> PlanatSole(n,3); # Peter Luschny, Oct 23 2011

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 *)

use ntheory ":all"; forprimes { say if 3 * (LogarithmicIntegral(chebyshev_psi($**3)) - LogarithmicIntegral(chebyshev_psi($**3-1))) < 1 } 3, 1000; # Dana Jacobsen, Dec 29 2015

def A196669(n) : return PlanatSole(n,3)
# The function PlanatSole(n,r) is in A196667.
# Peter Luschny, Oct 23 2011

Corrected and extended by Dana Jacobsen, Dec 29 2015

A196670 The Chebyshev primes of index 4.

Original entry on oeis.org

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

Offset: 1

Michel Planat, Oct 05 2011

nonn

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.

Dana Jacobsen, Table of n, a(n) for n = 1..75
M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT], 2011.

Cf. A196667, A196668, A196669.

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;

# The function PlanatSole(n,r) is in A196667.
A196670 := n -> PlanatSole(n,4); # Peter Luschny, Oct 23 2011

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 *)

use ntheory ":all"; forprimes { say if 4 *(LogarithmicIntegral(chebyshev_psi($**4)) - LogarithmicIntegral(chebyshev_psi($**4-1))) < 1 } 3,100; # Dana Jacobsen, Dec 29 2015

def A196670(n) : return PlanatSole(n,4)
# The function PlanatSole(n,r) is in A196667.
# Peter Luschny, Oct 23 2011

More terms from Dana Jacobsen, Dec 29 2015

A197186 The Riemann primes of the psi type and index 2.

Original entry on oeis.org

2, 17, 31, 41, 53, 101, 109, 127, 139, 179, 397, 419, 547, 787, 997, 1031, 1229, 1801, 1811, 2099, 2237, 2417, 2423, 2657, 3163, 3203, 3517, 3581, 3583, 3931, 4241, 5503, 5507, 5557, 6079, 8087, 8719, 10433, 10487, 13399, 13411, 19309, 22303, 22307, 22613

Offset: 1

Michel Planat, Oct 11 2011

nonn

The sequence consists of the prime numbers p that are champions (left to right maxima) of the function |psi(p^2)-p^2|, where psi(p) is the Chebyshev psi function.

Dana Jacobsen, Table of n, a(n) for n = 1..84
M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes, arXiv:1109.6489 [math.NT], 2011.

Cf. A196668, A197185, A197187, A197188.

ChebyshevPsi[n_] := Range[n] // MangoldtLambda // Total;
Reap[For[max=0; p=2, p < 2000, p = NextPrime[p], f = Abs[ChebyshevPsi[p^2] - p^2]; If[f > max, max = f; Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 03 2018 *)

use ntheory ":all"; my($max,$f)=(0); forprimes { $f=abs(chebyshev_psi($**2)-$**2); if ($f > $max) { say; $max=$f; } } 10000; # Dana Jacobsen, Dec 29 2015

More terms from Dana Jacobsen, Dec 29 2015

A197298 The Riemann primes of the theta type and index 2.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 59, 73, 97, 107, 109, 139, 179, 233, 263, 277, 283, 337, 347, 409, 419, 547, 643, 683, 809, 811, 821, 823, 863, 983, 991, 997, 1031, 1193

Offset: 1

Michel Planat, Oct 13 2011

nonn

The sequence consists of the prime numbers p that are champions (left to right maxima) of the function |theta(p^2)-p^2|, where theta(p) is the Chebyshev theta function.

Dana Jacobsen, Table of n, a(n) for n = 1..173
M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT], 2011.

Cf. A196668, A197186, A197297, A197299, A197300.

use ntheory ":all"; my($max,$f)=(0); forprimes { $f=abs(chebyshev_theta($**2)-$**2); if ($f > $max) { say; $max=$f; } } 10000; # Dana Jacobsen, Dec 29 2015

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A196667 The Chebyshev primes of index 1.

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A196669 The Chebyshev primes of index 3.

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A196670 The Chebyshev primes of index 4.

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A197186 The Riemann primes of the psi type and index 2.

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A197298 The Riemann primes of the theta type and index 2.

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Perl