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 1-3 of 3 results.

A230192 Decimal expansion of log(6^9*10^5)/25.

Original entry on oeis.org

1, 1, 0, 5, 5, 5, 0, 4, 2, 7, 5, 2, 0, 9, 0, 8, 9, 3, 7, 0, 9, 6, 0, 9, 0, 1, 3, 9, 9, 5, 3, 9, 2, 5, 6, 5, 9, 7, 0, 0, 4, 9, 6, 9, 4, 6, 9, 1, 1, 6, 3, 6, 2, 8, 9, 3, 1, 4, 6, 0, 0, 3, 4, 3, 7, 2, 0, 6, 3, 4, 1, 7, 1, 4, 0, 3, 2, 5, 9, 8, 2, 1, 7, 3, 9, 8, 1, 1, 9, 1, 0, 4, 6, 9, 5, 7, 3, 9, 3, 9, 1, 4, 7, 1, 8
Offset: 1

Views

Author

Arkadiusz Wesolowski, Oct 11 2013

Keywords

Comments

The value is equal to 6/5*(log(2)/2 + log(3)/3 + log(5)/5 - log(30)/30) = (6/5)*A230191.
Pafnuty Chebyshev proved in 1852 that A*x/log(x) < pi(x) < B*x/log(x) holds for all x >= x(0) with some x(0) sufficiently large, where A = 5/6*B and B is the constant given above.

Examples

			1.105550427520908937096090139953925659700496946911636289314600343720634...

References

  • Harold M. Edwards, Riemann's zeta function, Dover Publications, Inc., New York, 2001, pp. 281-284.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 164.

Links

Crossrefs

Cf. A000040, A000720, A230191.

Programs

  • Mathematica
    RealDigits[Log[6^9 10^5]/25,10,120][[1]] (* Harvey P. Dale, Mar 14 2015 *)
  • PARI
    default(realprecision, 105); x=log(6^9*10^5)/25; for(n=1, 105, d=floor(x); x=(x-d)*10; print1(d, ", "));

A323458 Decimal expansion of log(2^(1/2)*3^(1/3) / 6^(1/6)).

Original entry on oeis.org

4, 1, 4, 1, 5, 1, 1, 0, 8, 2, 9, 8, 0, 0, 0, 0, 5, 1, 7, 0, 4, 9, 5, 1, 5, 7, 9, 9, 7, 3, 1, 4, 6, 4, 7, 3, 4, 6, 6, 4, 1, 5, 1, 3, 7, 7, 5, 7, 2, 0, 9, 9, 9, 3, 3, 2, 9, 3, 4, 2, 3, 9, 2, 1, 0, 4, 0, 4, 6, 9, 2, 2, 8, 5, 9, 6, 6, 6, 3, 9, 9, 6, 8, 0, 8, 9, 0, 4, 0, 1, 4, 6, 7, 7, 6, 1, 5, 7, 7, 3
Offset: 0

Views

Author

N. J. A. Sloane, Jan 20 2019

Keywords

Examples

			0.4141511082980000517049515799731464734664151377572...

Links

Crossrefs

Suggested by A230191.
Cf. A002162, A002391, A016635.
Cf. A001008, A002805.

Programs

  • Mathematica
    RealDigits[Log[2^(1/2)*3^(1/3) / 6^(1/6)], 10, 101][[1]] (* Georg Fischer, Apr 04 2020 *)
  • PARI
    log( 2^(1/2)*3^(1/3) / 6^(1/6) ) \\ Charles R Greathouse IV, May 15 2019

Formula

From Jianing Song, Jan 23 2019: (Start)
Equals (1/6)*log(12) = (1/6)*A016635.
Equals (1/3)*log(2) + (1/6)*log(3) = (1/3)*A002162 + (1/6)*A002391. (End)
Equals Sum_{k>=1} H(2*k-1)/4^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, May 30 2021

Extensions

a(99) corrected by Georg Fischer, Apr 04 2020

A323459 Decimal expansion of log( 2^(1/2)*3^(1/3)*5^(1/5)*7^(1/7) / 210^(1/210) ).

Original entry on oeis.org

1, 2, 8, 7, 1, 9, 0, 0, 1, 6, 3, 2, 7, 3, 1, 4, 6, 7, 6, 6, 3, 0, 2, 4, 0, 1, 1, 5, 2, 4, 4, 9, 0, 7, 3, 4, 3, 2, 4, 0, 7, 3, 3, 2, 1, 4, 9, 8, 1, 5, 2, 1, 2, 3, 6, 4, 8, 5, 6, 4, 5, 2, 7, 5, 7, 4, 7, 2, 9, 1, 4, 1, 6, 3, 4, 6, 4, 5, 8, 3, 1, 5, 4, 9, 0, 3, 1, 8, 3, 7, 4, 0, 5, 0, 6, 8, 4, 2, 6, 6, 4
Offset: 1

Views

Author

N. J. A. Sloane, Jan 20 2019

Keywords

Examples

			1.28719001632731467663024011524490734324073321498152...

Links

Crossrefs

Suggested by A230191.

Programs

  • Mathematica
    RealDigits[Log[( 2^(1/2)*3^(1/3)*5^(1/5)*7^(1/7) / 210^(1/210) )], 10, 101] (* Georg Fischer, Apr 04 2020 *)
  • PARI
    log( 2^(1/2)*3^(1/3)*5^(1/5)*7^(1/7) / 210^(1/210) ) \\ Charles R Greathouse IV, May 15 2019

Extensions

a(100) corrected by Georg Fischer, Apr 04 2020
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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