A197187 -id:A197187 - cp's OEIS frontend

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

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

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

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

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

A196184 Positive integers b for which there is a 9-Pythagorean triple (a,b,c) satisfying a<=b.

Original entry on oeis.org

15, 30, 5, 8, 13, 45, 160, 60, 24, 48, 75, 459, 10, 16, 26, 90, 320, 57, 105, 15, 45, 120, 273, 15, 24, 39, 104, 135, 184, 480, 48, 96, 150, 165, 285, 20, 32, 52, 180, 640, 195, 240, 408, 114, 210, 17, 25, 40, 65, 72, 112, 144, 225, 329, 416, 553, 800, 21

Offset: 1

Clark Kimberling, Sep 29 2011

nonn

See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.

Cf. A195770, A196183, A196185, A196186, A197187.

Mathematica
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(See A196183.)
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cp's OEIS Frontend

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

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

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

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

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A196184 Positive integers b for which there is a 9-Pythagorean triple (a,b,c) satisfying a<=b.

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Mathematica