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.

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A215755 Denominators of the continued fraction convergents of log_10(7).

Original entry on oeis.org

1, 1, 6, 13, 71, 439, 510, 2455069, 2455579, 4910648, 12276875, 29464398, 71205671, 100670069, 171875740, 272545809, 1807150594, 9308298779, 11115449373, 931890596738, 6534349626539, 14000589849816, 20534939476355, 34535529326171, 55070468802526, 475099279746379, 2430566867534421
Offset: 0

Views

Author

V. Raman, Aug 23 2012

Keywords

Comments

7^a(n) gets increasingly close to 10^(numerator of convergent).

Crossrefs

Numerators are in A215759.
Cf. A046104, A073733, A082571, A153620.

Programs

  • Mathematica
    Rest[Denominator[Convergents[Log[10,7],30]]] (* Harvey P. Dale, Aug 23 2013 *)
  • PARI
    {my(cf=contfrac(log(7)/log(10))); vector(#cf, i, contfracpnqn( cf[1..i])[2, 1])}

Extensions

a(0)=1 prepended by Andrew Howroyd, Jul 09 2024

A215759 Numerators of the continued fraction convergents of log_10(7).

Original entry on oeis.org

0, 1, 5, 11, 60, 371, 431, 2074774, 2075205, 4149979, 10375163, 24900305, 60175773, 85076078, 145251851, 230327929, 1527219425, 7866425054, 9393644479, 787538916811, 5522166062156, 11831871041123, 17354037103279, 29185908144402, 46539945247681, 401505470125850, 2054067295876931
Offset: 0

Views

Author

V. Raman, Aug 23 2012

Keywords

Comments

7^(denominator of convergent) gets increasingly close to 10^a(n), agreeing to approximately a(n) digits

Crossrefs

Denominators are in A215755.
Cf. A046104, A073733, A082571, A153620.

Programs

  • Mathematica
    Rest[Numerator[Convergents[Log[10,7],30]]] (* Harvey P. Dale, Feb 16 2014 *)
  • PARI
    {my(cf=contfrac(log(7)/log(10))); vector(#cf, i, contfracpnqn( cf[1..i])[1, 1])}

Extensions

a(0)=0 prepended by Andrew Howroyd, Jul 09 2024

A333332 Positive numbers k at which min{abs(2^k - 10^y)/10^y: y in Z} reaches a new minimum.

Original entry on oeis.org

1, 2, 3, 10, 93, 196, 485, 2136, 13301, 28738, 42039, 70777, 254370, 325147, 6107016, 6432163, 44699994, 51132157, 146964308, 198096465, 345060773, 1578339557, 1923400330, 82361153417, 496090320832, 578451474249, 2809896217828, 6198243909905, 21404627947543
Offset: 1

Views

Author

Zachary Hervieux-Moore, Mar 15 2020

Keywords

Comments

If {k(n)/y(n)} are the convergent fractions to log_2(10), then numerators k(n) are in A073733, and denominators y(n) are in A046104; now, k and y means k(n) and y(n): k/y ~ log_2(10) <==> 2^(k/y) ~ 10 <==> 2^k ~ 10^y <==> lim_{n->oo} (2^k / 10^y) = 1 <==> lim_{n->oo} abs(2^k/10^y - 1) = 0 <==> lim_{n->oo} abs(2^k - 10^y)/10^y = 0, that corresponds to the name. - Bernard Schott, Apr 29 2020

Crossrefs

Cf. A046104, A073733.

Programs

  • Python
    def closest_powers_of_2_to_10(n):
  smallest_error = 1
  a = []
  r = 0.2 # ratio test starts at 2/10
  k = 1
  while len(a) < n:
    error = abs(1-r)
    if error < smallest_error:
      smallest_error = error
      a.append(k)
      print(a)
    if r<1.0:
      r *= 2
    else:
      r /= 10
      k -= 1 # need to check the other power of 10
    k += 1
  return a
print(closest_powers_of_2_to_10(20))

Extensions

More terms from Hugo Pfoertner, May 01 2020
Previous Showing 11-13 of 13 results.

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