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

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

Author

Zachary Hervieux-Moore, Mar 15 2020

Keywords

• nonn

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