Michel Planat - cp's OEIS frontend

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

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

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^4)-p^4|, where theta(p) is the Chebyshev theta function, theta(x) = sum_{primes p <=x } log p.

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

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

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

More terms from Dana Jacobsen, Dec 28 2015

A197299 The Riemann primes of the theta type and index 3.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 31, 37, 41, 43, 47, 53, 59, 67, 73, 83, 89, 101, 103, 137, 163, 167, 179, 197, 211, 223, 239, 251, 277, 331, 379, 397, 431, 463, 467, 521, 577, 593, 601, 613, 617, 719, 809, 881, 919, 967, 1091, 1123, 1129, 1237, 1249, 1289

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^3)-p^3|, where theta(p) is the Chebyshev theta function.

Dana Jacobsen, Table of n, a(n) for n = 1..85
M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT]

Cf. A196669, A197187, A197297, A197298, A197300.

use ntheory ":all"; my($max,$f)=(0); forprimes { $f=abs(chebyshev_theta($**3)-$**3); if ($f > $max) { say; $max=$f; } } 1000; # Dana Jacobsen, Dec 28 2015

More terms from Dana Jacobsen, Dec 28 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

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

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

Michel Planat, Oct 11 2011

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.

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

Cf. A196670, A197185, A197186, A197187.

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

More terms from Dana Jacobsen, Dec 28 2015

A197187 The Riemann primes of the psi type and index 3.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 29, 59, 67, 97, 103, 149, 151, 233, 251, 277, 311, 313, 479, 643, 719, 919, 967, 1039, 1373, 1489, 1553, 1847, 1973, 1979, 2663, 2953, 3323, 3677, 3691, 4651, 4663, 4789

Offset: 1

Michel Planat, Oct 11 2011

The sequence consists of the prime numbers p that are champions (left to right maxima) of the function |psi(p^3)-p^3|, where psi(p) is the Chebyshev psi function.

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

Cf. A196669, A197185, A197186, A197188.

ChebyshevPsi[n_] := Range[n] // MangoldtLambda // Total;
Reap[For[max=0; p=2, p < 1000, p = NextPrime[p], f = Abs[ChebyshevPsi[p^3] - p^3]; 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 { $f=abs(chebyshev_psi($**3)-$**3); if ($f > $max) { say; $max=$f; } } 1000; # Dana Jacobsen, Dec 28 2015

More terms from Dana Jacobsen, Dec 28 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

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

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).

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]

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User: Michel Planat

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

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A197299 The Riemann primes of the theta type and index 3.

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

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

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

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A197187 The Riemann primes of the psi type and index 3.

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

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

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