A363060 Numbers k such that 5 is the first digit of 2^k.

9, 19, 29, 39, 49, 59, 69, 102, 112, 122, 132, 142, 152, 162, 172, 195, 205, 215, 225, 235, 245, 255, 265, 298, 308, 318, 328, 338, 348, 358, 391, 401, 411, 421, 431, 441, 451, 461, 494, 504, 514, 524, 534, 544, 554, 587, 597, 607, 617, 627, 637, 647, 657, 680, 690

Offset: 1

Author

Ctibor O. Zizka, May 16 2023

Keywords

• nonn
• base
• easy

Comments

The asymptotic density of this sequence is log_10(6/5) = 0.0791812... . - Amiram Eldar, May 16 2023

In base B we may consider numbers k such that some integer Y >= 1 forms the first digit(s) of X^k. For such numbers k the following inequality holds: log_B(Y) - floor(log_B(Y)) <= k*log_B(X) - floor(k*log_B(X)) < log_B(Y+1) - floor(log_B(Y+1)). The irrationality of log_B(X) is the necessary condition; see the Links section. Examples in the OEIS: B = 10, X = 2; Y = 1 (A067497), Y = 2 (A067469), Y = 3 (A172404).

Examples

			k = 9: the first digit of 2^9 = 512 is 5, thus k = 9 is a term.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• P. Bohl, Über ein in der Theorie der säkularen Störungen vorkommendes Problem, J. Reine Angew. Math. 135 (1909), 189-283.
• Luboš Pick, On One Application of Irrationality in Search of the Seventh Heaven, Pokroky matematiky, fyziky a astronomie, vol. 67 (2022), 37-44.
• Hermann Weyl, Über die gibbs’sche Erscheinung und verwandte Konvergenzphänomene, Rend. Circ. Matem. Palermo 30 (1910), 377-407.
• Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins., Mathematische Annalen 77 (1916), 313-352.

Crossrefs

Cf. A000079, A008952, A018856.

Cf. A067497, A067469, A172404, A367294, A367295, A330243, A367296, A097415.

Programs

Maple

R:= NULL: count:= 0: t:= 1:
for k from 1 while count < 100 do
  t:= 2*t;
  if floor(t/10^ilog10(t)) = 5 then R:= R,k; count:= count+1 fi
od:
R; # Robert Israel, May 19 2023

Mathematica

Select[Range[700], IntegerDigits[2^#][[1]] == 5 &] (* Amiram Eldar, May 16 2023 *)

PARI

isok(k) = digits(2^k)[1] == 5; \\ Michel Marcus, May 16 2023

Python

from itertools import count, islice
def A363060_gen(startvalue=1): # generator of terms >= startvalue
    m = 1<<(k:=max(startvalue,1))
    for i in count(k):
        if str(m)[0]=='5':
            yield i
        m <<= 1
A363060_list = list(islice(A363060_gen(),20)) # Chai Wah Wu, May 21 2023