A196667 -id:A196667 - cp's OEIS frontend

A196668 The Chebyshev primes of index 2.

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

Michel Planat, Oct 05 2011

nonn

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.

Dana Jacobsen, Table of n, a(n) for n = 1..4035
M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT], 2011.

Cf. A196667, A196669, 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;
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;

# The function PlanatSole(n,r) is in A196667.
A196668 := n -> PlanatSole(n,2); # Peter Luschny, Oct 23 2011

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 *)

use ntheory ":all"; forprimes { say if 2 * (LogarithmicIntegral(chebyshev_psi($**2)) - LogarithmicIntegral(chebyshev_psi($**2-1))) < 1 } 3, 1000; # Dana Jacobsen, Dec 29 2015

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

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

A197185 The Riemann primes of the psi type and index 1.

Original entry on oeis.org

2, 59, 73, 97, 109, 113, 199, 283, 463, 467, 661, 1103, 1109, 1123, 1129, 1321, 1327, 1423, 2657, 2803, 2861, 3299, 5381, 5881, 6373, 6379, 9859, 9931, 9949, 10337, 10343, 11777, 19181, 19207, 19373, 24107, 24109, 24113, 24121, 24137, 42751, 42793, 42797

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)-p|, where psi(p) is the Chebyshev psi function.

Dana Jacobsen, Table of n, a(n) for n = 1..583
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. A196667, A197186, A197187, A197188.

ChebyshevPsi[n_] := Range[n] // MangoldtLambda // Total;
Reap[For[max=0; p=2, p<50000, p = NextPrime[p], f = Abs[ChebyshevPsi[p]-p]; 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 { $f=abs(chebyshev_psi($)-$); if ($f > $max) { say; $max=$f; } } 10000; # Dana Jacobsen, Dec 29 2015

A197297 The Riemann primes of the theta type and index 1.

Original entry on oeis.org

2, 5, 7, 11, 17, 29, 37, 41, 53, 59, 97, 127, 137, 149, 191, 223, 307, 331, 337, 347, 419, 541, 557, 809, 967, 1009, 1213, 1277, 1399, 1409, 1423, 1973, 2203, 2237, 2591, 2609, 2617, 2633, 2647, 2657, 3163, 3299, 4861, 4871, 4889, 4903, 4931, 5381, 7411

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)-p|, where theta(p) is the Chebyshev theta function.

Dana Jacobsen, Table of n, a(n) for n = 1..744
Michel Planat and Patrick 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 (1976) 337-360.

Cf. A028388, A196667, A197185, A215013.

Equivalent sequences for other indices: A197298(2), A197299(3), A197300(4).

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

A196675 The n-th Chebyshev primes that are equal to the 2n-th primes.

Original entry on oeis.org

164051, 231299, 255919, 274177, 343517, 447827, 450451, 528167, 587519, 847607, 1209469

Offset: 1

Michel Planat, Oct 06 2011

nonn

The Chebyshev primes (of index 1) are such odd primes that satisfy li[psi(p)]-li[psi(p-1)]<1 (sequence A196667), where li(x) is the logarithmic integral and psi(x) is the Chebyshev's psi function.
The present sequence lists the n-th Chebyshev primes that are equal to the 2n-th primes.
See A196674 for the number n at the zeros of the function a(n)-p_{2n}, where a(n) is the n-th Chebyshev prime and p_{2n} the 2n-th prime.

M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT], 2011.

Cf. A196674 (number n at the zeros of the function a(n)-p_{2n}, where a(n) is the n-th Chebyshev prime and p_{2n} the 2n-th prime).

A196673 Chebyshev primes that begin a record gap to the next Chebyshev prime.

Original entry on oeis.org

109, 113, 139, 317, 887, 1327, 1913, 3089, 8297, 11177, 29761, 45707, 113383, 164893, 291377, 401417, 638371, 1045841

Offset: 1

Michel Planat, Oct 05 2011

nonn

The Chebyshev primes (of index 1) are such odd primes that satisfy li[psi(p)]-li[psi(p-1)]<1 (sequence A196667), where li(x) is the logarithmic integral and psi(x) is the Chebyshev's psi function.
The present sequence lists the Chebyshev primes that begin a record gap to the next Chebyshev prime.
See A196672 for the length of the gap.

M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT]

Cf. A002386 (primes beginning a record gap).

Cf. A182876 (Ramanujan primes that begin a record gap to the next Ramanujan prime).

A196674 Numbers n such that c(n) = p_{2n}, where c(n) is the n-th Chebyshev prime and p_{2n} the 2n-th prime.

Original entry on oeis.org

510, 10271, 11259, 11987, 14730, 18772, 18884, 21845, 24083, 33723, 46789

Offset: 1

Michel Planat, Oct 05 2011

nonn

The Chebyshev primes (of index 1) are such odd primes that satisfy li[psi(p)]-li[psi(p-1)]<1 (sequence A196667), where li(x) is the logarithmic integral and psi(x) is the Chebyshev's psi function.
THe present sequence lists the zeros of the function c(n)-p_{2n}, where c(n) is the n-th Chebyshev prime and p_{2n} the 2n-th prime.
See A196675 for the Chebyshev primes satisfying a(n)=p_{2n}.

M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT]

A196671 The number of Chebyshev primes (of index 1) less than 10^n, (n=1,2,...).

Original entry on oeis.org

0, 0, 42, 516, 4498, 414123

Offset: 1

Michel Planat, Oct 05 2011

nonn

The Chebyshev primes (of index 1) are such odd primes that satisfy li[psi(p)]-li[psi(p-1)]<1 (sequence A196667), where li(x) is the logarithmic integral and psi(x) is the Chebyshev's psi function.
The present sequence lists the number of Chebyshev primes less than 10^n, (n=1,2,...).
The sequence suggests the density pi(x)/2 for the Chebyshev primes, where pi(x) is the density of primes.

M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT]

Cf. A196667.

cp's OEIS Frontend

A196668 The Chebyshev primes of index 2.

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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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A197185 The Riemann primes of the psi type and index 1.

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A197297 The Riemann primes of the theta type and index 1.

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Perl

A196675 The n-th Chebyshev primes that are equal to the 2n-th primes.

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A196673 Chebyshev primes that begin a record gap to the next Chebyshev prime.

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A196674 Numbers n such that c(n) = p_{2n}, where c(n) is the n-th Chebyshev prime and p_{2n} the 2n-th prime.

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