A008952 -id:A008952 - cp's OEIS frontend

A364185 Leading digit of 11^n.

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6, 6, 7, 8, 8, 9, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 1, 1, 1, 1

Offset: 0

Seiichi Manyama, Jul 15 2023

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A008952, A060956, A111395, A362871, A363093, A363249.

Cf. A000030, A001020, A008571.

a[n_] := IntegerDigits[11^n][[1]]; Array[a, 100, 0] (* Amiram Eldar, Jul 15 2023 *)

PARI
```
a(n) = digits(11^n)[1];
```

a(n) = A000030(A001020(n)).

A000689 Final decimal digit of 2^n.

Original entry on oeis.org

1, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6

Offset: 0

N. J. A. Sloane

These are the analogs of the powers of 2 in carryless arithmetic mod 10.
Let G = {2,4,8,6}. Let o be defined as XoY = least significant digit in XY. Then (G,o) is an Abelian group wherein 2 is a generator (also see the first comment under A001148). - K.V.Iyer, Mar 12 2010
This is also the decimal expansion of 227/1818. - Kritsada Moomuang, Dec 21 2021

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 6*x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 6*x^8 + ...

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version
Index entries for sequences related to carryless arithmetic
Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A000079, A000855, A008952, A034887, A173635.

a000689 n = a000689_list !! n
a000689_list = 1 : cycle [2,4,8,6]  -- Reinhard Zumkeller, Sep 15 2011

[2^n mod 10: n in [0..150]]; // Vincenzo Librandi, Apr 12 2011

Table[PowerMod[2, n, 10], {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)

for(n=0,80, if(n,{x=(n+3)%4+1; print1(10-(4*x^3+47*x-27*x^2)/3,", ")},{print1("1, ")}))

[power_mod(2,n,10)for n in range(0, 81)] # Zerinvary Lajos, Nov 03 2009

Periodic with period 4.
a(n) = 2^n mod 10.
a(n) = A002081(n) - A002081(n-1), for n > 0.
From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3), n > 3.
G.f.: (x+3*x^2+5*x^3+1)/((1-x) * (1+x^2)). (End)
For n >= 1, a(n) = 10 - (4x^3 + 47x - 27x^2)/3, where x = (n+3) mod 4 + 1.
For n >= 1, a(n) = A070402(n) + 5*floor( ((n-1) mod 4)/2 ).
G.f.: 1 / (1 - 2*x / (1 + 5*x^3 / (1 + x / (1 - 3*x / (1 + 3*x))))). - Michael Somos, May 12 2012
a(n) = 5 + cos((n*Pi)/2) - 3*sin((n*Pi)/2) for n >= 1. - Kritsada Moomuang, Dec 21 2021

A111395 First digit of powers of 5.

Original entry on oeis.org

1, 5, 2, 1, 6, 3, 1, 7, 3, 1, 9, 4, 2, 1, 6, 3, 1, 7, 3, 1, 9, 4, 2, 1, 5, 2, 1, 7, 3, 1, 9, 4, 2, 1, 5, 2, 1, 7, 3, 1, 9, 4, 2, 1, 5, 2, 1, 7, 3, 1, 8, 4, 2, 1, 5, 2, 1, 6, 3, 1, 8, 4, 2, 1, 5, 2, 1, 6, 3, 1, 8, 4, 2, 1, 5, 2, 1, 6, 3, 1, 8, 4, 2, 1, 5, 2, 1, 6

Offset: 0

Almerio A. Castro (almerio.castro(AT)gmail.com), Nov 11 2005

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A008952, A060956.

Cf. A000030, A000351, A008566.

First[IntegerDigits[#]]&/@(5^Range[0,100]) (* Harvey P. Dale, Jan 13 2015 *)

a(n) = digits(5^n)[1]; \\ Michel Marcus, Jan 07 2014

a(n) = A000030(A000351(n)). - Seiichi Manyama, Jul 15 2023

a(0)=1 prepended, and more terms from Michel Marcus, Jan 07 2014

A320859 Powers of 2 with initial digit 3.

Original entry on oeis.org

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).

A320860 Powers of 2 with initial digit 4.

Original entry on oeis.org

4, 4096, 4194304, 4294967296, 4398046511104, 4503599627370496, 4611686018427387904, 4722366482869645213696, 4835703278458516698824704, 4951760157141521099596496896, 40564819207303340847894502572032, 41538374868278621028243970633760768

Offset: 1

Muniru A Asiru, Oct 22 2018

Differs from A067482 first at n = 11.

Muniru A Asiru, Table of n, a(n) for n = 1..320
Index to divisibility sequences

Cf. A000079 (Powers of 2), A008952 (leading digit of 2^n), A217397 (numbers starting with 4).

Powers of 2 with initial digit k, (k = 1..4): A067488, A067480, A320859, this sequence.

Filtered(List([0..150],n->2^n),i->ListOfDigits(i)[1]=4);

[2^n: n in [1..160] | Intseq(2^n)[#Intseq(2^n)] eq 4]; // G. C. Greubel, Oct 27 2018

select(x->"4"=""||x[1],[2^n$n=0..150])[];

Select[2^Range[160], First[IntegerDigits[#]] == 4 &] (* G. C. Greubel, Oct 27 2018 *)

select(x->(digits(x)[1]==4), vector(200, n, 2^n)) \\ Michel Marcus, Oct 26 2018

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

A320861 Powers of 2 with initial digit 5.

Original entry on oeis.org

512, 524288, 536870912, 549755813888, 562949953421312, 576460752303423488, 590295810358705651712, 5070602400912917605986812821504, 5192296858534827628530496329220096, 5316911983139663491615228241121378304, 5444517870735015415413993718908291383296

Offset: 1

Muniru A Asiru, Oct 23 2018

Robert Israel, Table of n, a(n) for n = 1..263
Index to divisibility sequences

Cf. A000079 (powers of 2), A008952 (leading digit of 2^n).

Powers of 2 with initial digit k, (k = 1..5): A067488, A067480, A320859, A320860, this sequence.

Filtered(List([0..160],n->2^n),i->ListOfDigits(i)[1]=5);

[2^n: n in [1..200] | Intseq(2^n)[#Intseq(2^n)] eq 5]; // Vincenzo Librandi, Oct 25 2018

select(x->"5"=""||x[1],[2^n$n=0..160])[];
# Alternative:
Res:= NULL: count:= 0:
for k from 1 to 49 do
   n:= ilog2(6*10^k);
   if n > ilog2(5*10^k) then count:= count+1;
     Res:= Res, 2^n;
   fi
od:
Res; # Robert Israel, Oct 26 2018

Select[2^Range[200], First[IntegerDigits[#]]==5 &] (* Vincenzo Librandi, Oct 25 2018 *)

lista(nn) = {for(n=1, nn, x = 2^n; if (digits(x=2^n)[1] == 5, print1(x, ", ")););} \\ Michel Marcus, Oct 25 2018

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

A320862 Powers of 2 with initial digit 6.

Original entry on oeis.org

64, 65536, 67108864, 68719476736, 604462909807314587353088, 618970019642690137449562112, 633825300114114700748351602688, 649037107316853453566312041152512, 664613997892457936451903530140172288, 680564733841876926926749214863536422912

Offset: 1

Muniru A Asiru, Oct 23 2018

Muniru A Asiru, Table of n, a(n) for n = 1..220
Index to divisibility sequences

Cf. A000079 (powers of 2), A008952 (leading digit of 2^n), A217399 (numbers starting with 6).

Powers of 2 with initial digit k, (k = 1..6): A067488, A067480, A320859, A320860, A320861, this sequence.

Filtered(List([0..180],n->2^n),i->ListOfDigits(i)[1]=6);

[2^n: n in [1..160] | Intseq(2^n)[#Intseq(2^n)] eq 6]; // G. C. Greubel, Oct 27 2018

select(x->"6"=""||x[1],[2^n$n=0..180])[];

Select[2^Range[160], First[IntegerDigits[#]] == 6 &] (* G. C. Greubel, Oct 27 2018 *)

select(x->(digits(x)[1]==6), vector(200, n, 2^n)) \\ Michel Marcus, Oct 26 2018

A363249 Leading digit of 9^n.

Original entry on oeis.org

1, 9, 8, 7, 6, 5, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 9, 8, 7, 7, 6, 5, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 9, 8, 7, 7, 6, 5, 5, 4, 4, 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 9, 8, 7, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 9, 8, 7, 6, 6, 5, 4, 4, 4, 3, 3, 2, 2

Offset: 0

Seiichi Manyama, Jul 15 2023

He, Xinwei; Hildebrand, A J; Li, Yuchen; Zhang, Yunyi, Complexity of Leading Digit Sequences, Discrete Mathematics and Theoretical Computer Science; 22 (2020), #14.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A008952, A060956, A111395, A362871, A363093, A364185.

Cf. A000030, A001019, A008570.

a[n_] := IntegerDigits[9^n][[1]]; Array[a, 100, 0] (* Amiram Eldar, Jul 15 2023 *)

PARI
```
a(n) = digits(9^n)[1];
```

a(n) = A000030(A001019(n)).

a(n) = A060956(2*n).

cp's OEIS Frontend

A364185 Leading digit of 11^n.

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A000689 Final decimal digit of 2^n.

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A111395 First digit of powers of 5.

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A320859 Powers of 2 with initial digit 3.

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A320860 Powers of 2 with initial digit 4.

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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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A320861 Powers of 2 with initial digit 5.

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