cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A016040 Integer part of Chebyshev's theta function: floor( log(Product_{k=1..n} prime(k)) ).

Original entry on oeis.org

0, 1, 3, 5, 7, 10, 13, 16, 19, 22, 26, 29, 33, 37, 40, 44, 49, 53, 57, 61, 65, 70, 74, 79, 83, 88, 92, 97, 102, 107, 111, 116, 121, 126, 131, 136, 141, 146, 151, 157, 162, 167, 172, 177, 183, 188, 193, 199, 204, 210, 215, 221, 226, 232, 237, 243, 248
Offset: 1

Views

Author

Robert G. Wilson v

Keywords

Links

Crossrefs

Cf. A035158.

Programs

  • Mathematica
    Table[Floor[N[Sum[Log[Prime[x]], {x, 1, n}]]], {n, 1, 1000}] (* Artur Jasinski, Jan 23 2007 *)

Formula

a(n) = A000195(A002110(n)).
a(n) ~ n log n by the prime number theorem. - Charles R Greathouse IV, Dec 11 2008

Extensions

New name from Charles R Greathouse IV, Dec 11 2008

A057872 A version of the Chebyshev function theta(n): a(n) = round(Sum_{primes p <= n } log(p)).

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 3, 5, 5, 5, 5, 8, 8, 10, 10, 10, 10, 13, 13, 16, 16, 16, 16, 19, 19, 19, 19, 19, 19, 23, 23, 26, 26, 26, 26, 26, 26, 30, 30, 30, 30, 33, 33, 37, 37, 37, 37, 41, 41, 41, 41, 41, 41, 45, 45, 45, 45, 45, 45, 49, 49, 53, 53, 53, 53, 53, 53, 57, 57, 57, 57, 62, 62, 66, 66, 66, 66
Offset: 0

Views

Author

N. J. A. Sloane, Oct 02 2008

Keywords

Comments

See A035158, which is the main entry for this function.
The old entry with this sequence number was a duplicate of A053632.

References

  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 340.
  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.35, p. 267.

Links

Crossrefs

Cf. A034386, A215259, A215260.

Programs

  • PARI
    v=List(); t=0; for(n=0, 100, if(isprime(n), t+=log(n)); listput(v, round(t))); Vec(v) \\ Charles R Greathouse IV, Sep 23 2012

Formula

theta(n) = log(A034386(n)).
a(n) ~ n, a statement equivalent to the Prime Number Theorem. - Charles R Greathouse IV, Sep 23 2012

A083535 A version of the Chebyshev function theta(n): a(n) = ceiling(Sum_{primes p <= n } log(p)).

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 6, 6, 6, 6, 8, 8, 11, 11, 11, 11, 14, 14, 17, 17, 17, 17, 20, 20, 20, 20, 20, 20, 23, 23, 27, 27, 27, 27, 27, 27, 30, 30, 30, 30, 34, 34, 38, 38, 38, 38, 41, 41, 41, 41, 41, 41, 45, 45, 45, 45, 45, 45, 50, 50, 54, 54, 54, 54, 54, 54, 58, 58, 58, 58, 62, 62, 66, 66, 66, 66
Offset: 1

Views

Author

N. J. A. Sloane, Oct 02 2008

Keywords

Comments

See A035158, which is the main entry for this function.
The old entry with this sequence number was a duplicate of A083235.

Formula

a(n) ~ n by the prime number theorem. - Charles R Greathouse IV, Aug 02 2012

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

Views

Author

Michel Planat, Oct 13 2011

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Perl
    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

Extensions

More terms from Dana Jacobsen, Dec 28 2015

A252398 Successive n with minimal relative distance |1-theta(n)/n|, where theta(n) = log(A034386(n)) is Chebyshev's theta function.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 43, 47, 73, 103, 107, 109, 113, 199, 283, 467, 661, 887, 1063, 1069, 1097, 1103, 1109, 1123, 1129, 1303, 1307, 1321, 1327, 1621, 1627, 2803, 3931, 3947, 4273, 4289, 4297, 5867, 5869, 5881, 6373, 6379, 9439, 9473, 9479, 9497, 9551, 9859, 9931, 9949
Offset: 1

Views

Author

Jean-François Alcover, Dec 17 2014

Keywords

Comments

The first 10000 terms are the same as A108310 (see that sequence for comments). - Charles R Greathouse IV, Dec 18 2014
This sequence, unlike A108310, is presumably infinite; it is finite if and only if theta(n) = n for some number n.

Examples

			Given that 1 - theta(3)/3 = 1 - log(6)/3 = 0.40..., 1 - theta(4)/4 = 1 - log(6)/4 = 0.55... and 1 - theta(5)/5 = 1 - log(30)/5 = 0.31..., the next term after 3 is 5.

Links

Crossrefs

Cf. A016040, A035158, A057872, A083535, A108310, A215013.

Programs

  • Mathematica
    (* Adapted from PARI *) Reap[For[record = 2; theta = 0; p = 2, p < 2 * 10^8, p = NextPrime[p], theta = theta + Log[p] //N; d = Abs[1 - theta/p]; If[d < record, record = d; Print[p]; Sow[p]]]][[2, 1]]
  • PARI
    /* Note: This program may fail if you replace 1e6 with a number larger than 1e17, and will certainly fail at some point below 1e316. These large numbers are not remotely feasible at the moment. */
r=th=0; forprime(p=2,1e6, th+=log(p); t=th/p; if(t>r, r=t; print1(p", "); if(t>1, warning("theta(n) > n, possible missed terms")))) \\ Charles R Greathouse IV, Dec 17 2014
Showing 1-5 of 5 results.

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