A172404 -id:A172404 - cp's OEIS frontend

A320859 Powers of 2 with initial digit 3.

32, 32768, 33554432, 34359738368, 35184372088832, 36028797018963968, 36893488147419103232, 37778931862957161709568, 302231454903657293676544, 38685626227668133590597632, 309485009821345068724781056, 39614081257132168796771975168, 316912650057057350374175801344

Offset: 1

Muniru A Asiru, Oct 22 2018

Muniru A Asiru, Table of n, a(n) for n = 1..410
Index to divisibility sequences

Cf. A000079 (Powers of 2), A008952 (leading digit of 2^n).
Powers of 2 with initial digit k, (k = 1..4): A067488, A067480, this sequence, A320860.
Cf. A172404.

Filtered(List([0..120],n->2^n),i->ListOfDigits(i)[1]=3);

[2^n: n in [1..100] | Intseq(2^n)[#Intseq(2^n)] eq 3]; // G. C. Greubel, Oct 24 2018

select(x->"3"=""||x[1],[2^n$n=0..120])[];

Select[2^Range[0, 100], First[IntegerDigits[#]] == 3 &] (* G. C. Greubel, Oct 24 2018 *)

lista(nn) = {for(n=1, nn, x = 2^n; if (digits(x=2^n)[1] == 3, print1(x, ", ")););} \\ Michel Marcus, Oct 25 2018

a(n) = 2^A172404(n).

A330243 Numbers k such that the first digit of the decimal expansion of 2^k is 7.

Original entry on oeis.org

46, 56, 66, 76, 86, 96, 149, 159, 169, 179, 189, 242, 252, 262, 272, 282, 292, 345, 355, 365, 375, 385, 438, 448, 458, 468, 478, 488, 531, 541, 551, 561, 571, 581, 634, 644, 654, 664, 674, 727, 737, 747, 757, 767, 777, 830, 840, 850, 860, 870, 923, 933, 943, 953

Offset: 1

Eder Vanzei, Dec 06 2019

The asymptotic density of this sequence is log_10(8/7) = 0.057991... - Amiram Eldar, Jan 27 2021

			70368744177664 = 2^46.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000030, A000079, A008952, A067469, A067497, A172404.

Select[Range[1000], Floor[2^# / 10^(Floor[# * Log10[2]])] == 7 &] (* Amiram Eldar, Dec 07 2019 *)
Select[Range[1000],IntegerDigits[2^#][[1]]==7&] (* or *) Select[Range[ 1000],NumberDigit[2^#,IntegerLength[2^#]-1]==7&] (* Harvey P. Dale, Aug 10 2021 *)

A330243_list = [n for n in range(10**3) if str(2**n)[0] == '7'] # Chai Wah Wu, Dec 12 2019

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

A367296 Numbers k such that 8 is the first digit of 2^k.

Original entry on oeis.org

3, 13, 23, 33, 43, 106, 116, 126, 136, 146, 199, 209, 219, 229, 239, 302, 312, 322, 332, 342, 395, 405, 415, 425, 435, 498, 508, 518, 528, 538, 591, 601, 611, 621, 631, 684, 694, 704, 714, 724, 787, 797, 807, 817, 827, 880, 890, 900, 910, 920, 983, 993, 1003

Offset: 1

Martin Renner, Nov 12 2023

The asymptotic density of this sequence is log_10(9/8) = 0.051152...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000079, A067497, A067469, A172404, A367294, A363060, A367295, A330243, A097415.

x := 1:
L := []:
for n from 0 to 10^3 do
  if 8 <= x and x < 9 then
    L := [op(L), n]
  fi;
  x := 2*x;
  if x > 10 then
    x := (1/10)*x fi;
od:
L;
# alternative:
select(t -> floor(2^t/10^ilog10(2^t))=8, [$1..10^4]); # Robert Israel, Nov 12 2024

Select[Range[1010], IntegerDigits[2^#][[1]] == 8 &] (* Amiram Eldar, Nov 12 2023 *)

from itertools import islice
def A367296_gen(): # generator of terms
    a, b, c, l = 8, 9, 1, 0
    while True:
        if a<=c:
            if cA367296_list = list(islice(A367296_gen(),30)) # Chai Wah Wu, Nov 13 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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A320859 Powers of 2 with initial digit 3.

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A330243 Numbers k such that the first digit of the decimal expansion of 2^k is 7.

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

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A367296 Numbers k such that 8 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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