A367296 -id:A367296 - cp's OEIS frontend

A172404 Numbers k such that 3 is the first digit of 2^k.

5, 15, 25, 35, 45, 55, 65, 75, 78, 85, 88, 95, 98, 108, 118, 128, 138, 148, 158, 168, 178, 181, 188, 191, 201, 211, 221, 231, 241, 251, 261, 271, 274, 281, 284, 291, 294, 304, 314, 324, 334, 344, 354, 364, 367, 374, 377, 384, 387, 397, 407, 417, 427, 437, 447, 457, 467, 470, 477, 480, 487, 490, 500

Offset: 1

David Radcliffe, Nov 20 2010

The asymptotic density of this sequence is log_10(4/3) = 0.124938... - Amiram Eldar, Jan 27 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 4200 terms from Muniru A Asiru)

Cf. A000079.

Cf. A067497, A067469, A367294, A363060, A367295, A330243, A367296, A097415.

Filtered([1..500],n->ListOfDigits(2^n)[1]=3); # Muniru A Asiru, Oct 17 2018

x := 1.; L := []; for n from 0 to 500 do if 3 < x and x < 4 then L := [op(L), n] end if; x := 2*x; if x > 10 then x := (1/10)*x end if end do; L;

Select[Range[1000], IntegerDigits[2^#][[1]] == 3 &]

isok(n) = digits(2^n)[1] == 3; \\ Michel Marcus, Oct 18 2018

ans, x = [], 1.
for n in range(501):
    if 3 < x < 4: ans.append(n)
    x = x*2
    if x > 10: x = x / 10
print(ans)

from itertools import islice
def A172404_gen(): # generator of terms
    a, b, c, l = 3, 4, 1, 0
    while True:
        if a<=c:
            if cA172404_list = list(islice(A172404_gen(),30)) # Chai Wah Wu, Nov 13 2023

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

Original entry on oeis.org

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

Ctibor O. Zizka, May 16 2023

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).

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

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.

Cf. A000079, A008952, A018856.

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

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

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

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

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

A367294 Numbers k such that 4 is the first digit of 2^k.

Original entry on oeis.org

2, 12, 22, 32, 42, 52, 62, 72, 82, 92, 105, 115, 125, 135, 145, 155, 165, 175, 185, 198, 208, 218, 228, 238, 248, 258, 268, 278, 288, 301, 311, 321, 331, 341, 351, 361, 371, 381, 394, 404, 414, 424, 434, 444, 454, 464, 474, 484, 497, 507, 517, 527, 537, 547

Offset: 1

Martin Renner, Nov 12 2023

The asymptotic density of this sequence is log_10(5/4) = 0.096910...

Cf. A000079, A067497, A067469, A172404, A363060, A367295, A330243, A367296, A097415.

x := 1:
L := []:
for n from 0 to 10^3 do
  if 4 <= x and x < 5 then
    L := [op(L), n]
  fi;
  x := 2*x;
  if x > 10 then
    x := (1/10)*x fi;
od:
L;

Select[Range[550], IntegerDigits[2^#][[1]] == 4 &] (* Amiram Eldar, Nov 12 2023 *)

A367295 Numbers k such that 6 is the first digit of 2^k.

Original entry on oeis.org

6, 16, 26, 36, 79, 89, 99, 109, 119, 129, 139, 182, 192, 202, 212, 222, 232, 275, 285, 295, 305, 315, 325, 335, 368, 378, 388, 398, 408, 418, 428, 471, 481, 491, 501, 511, 521, 564, 574, 584, 594, 604, 614, 624, 667, 677, 687, 697, 707, 717, 760, 770, 780, 790

Offset: 1

Martin Renner, Nov 12 2023

The asymptotic density of this sequence is log_10(7/6) = 0.066946...

Cf. A000079, A067497, A067469, A172404, A367294, A363060, A330243, A367296, A097415.

x := 1:
L := []:
for n from 0 to 10^3 do
  if 6 <= x and x < 7 then
    L := [op(L), n]
  fi;
  x := 2*x;
  if x > 10 then
    x := (1/10)*x fi;
od:
L;

Select[Range[800], IntegerDigits[2^#][[1]] == 6 &] (* Amiram Eldar, Nov 12 2023 *)

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A172404 Numbers k such that 3 is the first digit of 2^k.

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

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A367294 Numbers k such that 4 is the first digit of 2^k.

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A367295 Numbers k such that 6 is the first digit of 2^k.

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