A197299 -id:A197299 - cp's OEIS frontend

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

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

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

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

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

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

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

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

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