A008952 -id:A008952 - cp's OEIS frontend

A320863 Powers of 2 with initial digit 7.

70368744177664, 72057594037927936, 73786976294838206464, 75557863725914323419136, 77371252455336267181195264, 79228162514264337593543950336, 713623846352979940529142984724747568191373312, 730750818665451459101842416358141509827966271488

Offset: 1

Muniru A Asiru, Oct 26 2018

Muniru A Asiru, Table of n, a(n) for n = 1..170
Index to divisibility sequences

Cf. A000079 (powers of 2), A008952 (leading digit of 2^n), A217400 (numbers starting with 7).

Powers of 2 with initial digit k, (k = 1..7): A067488, A067480, A320859, A320860, A320861, A320862, this sequence.

Filtered(List([0..180],n->2^n),i->ListOfDigits(i)[1]=7);

[2^n: n in [1..160] | Intseq(2^n)[#Intseq(2^n)] eq 7]; // G. C. Greubel, Oct 27 2018

select(x->"7"=""||x[1],[2^n$n=0..180])[];

Select[2^Range[160], First[IntegerDigits[#]] == 7 &] (* G. C. Greubel, Oct 27 2018 *)

select(x->(digits(x)[1]==7), vector(200, n, 2^n)) \\ Michel Marcus, Oct 27 2018

A362871 Leading digit of 6^n.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jul 15 2023

Cf. A008952, A060956, A111395, A363093, A363249, A364185.

Cf. A000030, A000400, A008567.

a[n_] := IntegerDigits[6^n][[1]]; Array[a, 100, 0] (* Amiram Eldar, Jul 15 2023 *)

PARI
```
a(n) = digits(6^n)[1];
```

a(n) = A000030(A000400(n)).

A363093 Leading digit of 7^n.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jul 15 2023

Cf. A008952, A060956, A111395, A362871, A363249, A364185.

Cf. A000030, A000420, A001903, A008568.

a[n_] := IntegerDigits[7^n][[1]]; Array[a, 100, 0] (* Amiram Eldar, Jul 15 2023 *)

PARI
```
a(n) = digits(7^n)[1];
```

a(n) = A000030(A000420(n)).

A367361 Comma transform of powers of 2.

Original entry on oeis.org

12, 24, 48, 81, 63, 26, 41, 82, 65, 21, 42, 84, 68, 21, 43, 86, 61, 22, 45, 81, 62, 24, 48, 81, 63, 26, 41, 82, 65, 21, 42, 84, 68, 21, 43, 86, 61, 22, 45, 81, 62, 24, 48, 81, 63, 27, 41, 82, 65, 21, 42, 84, 69, 21, 43, 87, 61, 22, 45, 81, 62, 24, 49, 81, 63, 27, 41, 82, 65, 21, 42, 84, 69, 21, 43, 87, 61, 23, 46, 81

Offset: 0

N. J. A. Sloane, Nov 22 2023

See A367360 for further information.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000079, A000689, A008952, A166499, A367360.

FromDigits /@ Partition[Rest@ Flatten[{First[#], Last[#]} & /@ IntegerDigits[2^Range[0, 120]]], 2, 2] (* Michael De Vlieger, Nov 22 2023 *)

from itertools import count, islice, pairwise
def S(): yield from (str(2**i) for i in count(0))
def agen(): yield from (int(t[-1]+u[0]) for t, u in pairwise(S()))
print(list(islice(agen(), 80))) # Michael S. Branicky, Nov 22 2023

def A367361(n): return (60,20,40,80)[n&3]+int(str(1<Chai Wah Wu, Dec 22 2023

a(n) = 10 * A000689(n) + A008952(n+1). - Alois P. Heinz, Nov 22 2023

A141053 Most-significant decimal digit of Fibonacci(5n+3).

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Aug 01 2008

Leading digit of A134490(n).
From Johannes W. Meijer, Jul 06 2011: (Start)
The leading digit d, 1 <= d <= 9, of A141053 follows Benford’s Law. This law states that the probability for the leading digit is p(d) = log_10(1+1/d), see the examples.
We observe that the last digit of A134490(n), i.e. F(5*n+3) mod 10, leads to the Lucas sequence A000032(n) (mod 10), i.e. a repetitive sequence of 12 digits [2, 1, 3, 4, 7, 1, 8, 9, 7, 6, 3, 9] with p(0) = p(5) = 0, p(1) = p(3) = p(7) = p(9) = 1/6 and p(2) = p(4) = p(6) = p(8) = 1/12. This does not obey Benford’s Law, which would predict that the last digit would satisfy p(d) = 1/10, see the links. (End)

			From _Johannes W. Meijer_, Jul 06 2011: (Start)
d     p(N=2000) p(N=4000) p(N=6000) p(Benford)
1      0.29900   0.29950   0.30033   0.30103
2      0.17700   0.17675   0.17650   0.17609
3      0.12550   0.12525   0.12517   0.12494
4      0.09650   0.09675   0.09700   0.09691
5      0.07950   0.07950   0.07933   0.07918
6      0.06700   0.06675   0.06700   0.06695
7      0.05800   0.05825   0.05800   0.05799
8      0.05150   0.05125   0.05100   0.05115
9      0.04600   0.04600   0.04567   0.04576
Total  1.00000   1.00000   1.00000   1.00000 (End)

Hans J. H. Tuenter, Table of n, a(n) for n = 0..10000
Kevin Brown, Benford's Law.
Eric Weisstein's World of Mathematics, Benford's Law.
Wikipedia, Benford's Law.
Index entries for sequences related to Benford's law

Cf. A000045 (F(n)), A008963 (Initial digit F(n)), A105511-A105519, A003893 (F(n) mod 10), A130893, A186190 (First digit tribonacci), A008952 (Leading digit 2^n), A008905 (Leading digit n!), A045510, A112420 (Leading digit Collatz 3*n+1 starting with 1117065), A007524 (log_10(2)), A104140 (1-log_10(9)). - Johannes W. Meijer, Jul 06 2011

Cf. A001622, A097348.

A134490 := proc(n) combinat[fibonacci](5*n+3) ; end proc:
A141053 := proc(n) convert(A134490(n),base,10) ; op(-1,%) ; end proc:
seq(A141053(n),n=0..70) ; # R. J. Mathar, Jul 04 2011

Table[IntegerDigits[Fibonacci[5n+3]][[1]],{n,0,70}] (* Harvey P. Dale, Jun 22 2025 *)

a(n) = floor(F(5*n+3)/10^(floor(log(F(5*n+3))/log(10)))). - Johannes W. Meijer, Jul 06 2011

For n>0, a(n) = floor(10^{alpha*n+beta}), where alpha=5*log_10(phi)-1, beta=log_10(1+2/sqrt(5)), {x}=x-floor(x) denotes the fractional part of x, log_10(phi) = A097348, and phi = (1+sqrt(5))/2 = A001622. - Hans J. H. Tuenter, Aug 27 2025

Edited by Johannes W. Meijer, Jul 06 2011

A247243 a(n) = smallest positive integer k not already in the sequence such that the decimal expansion of 2^(n-1) begins with k.

Original entry on oeis.org

1, 2, 4, 8, 16, 3, 6, 12, 25, 5, 10, 20, 40, 81, 163, 32, 65, 13, 26, 52, 104, 209, 41, 83, 167, 33, 67, 134, 268, 53, 107, 21, 42, 85, 17, 34, 68, 137, 27, 54, 109, 219, 43, 87, 175, 35, 7, 14, 28, 56, 11, 22, 45, 9, 18, 36, 72, 144, 288, 57, 115, 23, 46, 92

Offset: 1

Paul Tek, Nov 30 2014

Is this a permutation of the positive integers?

			+----+---------+------+
+  n | 2^(n-1) | a(n) |
+----+---------+------+
|  1 |       1 |    1 |
|  2 |       2 |    2 |
|  3 |       4 |    4 |
|  4 |       8 |    8 |
|  5 |      16 |   16 |
|  6 |      32 |    3 |
|  7 |      64 |    6 |
|  8 |     128 |   12 |
|  9 |     256 |   25 |
| 10 |     512 |    5 |
| 11 |    1024 |   10 |
+----+---------+------+

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, Perl program for this sequence

Cf. A008952.

Perl
```
See Link section.
```

from itertools import count, islice
def ispal(n): s = str(n); return s == s[::-1]
def agen(): # generator of terms
    aset, mink = set(), 1
    for n in count(1):
        k, target = mink, str(2**(n-1))
        while k in aset or not target.startswith(str(k)): k += 1
        an = k; aset.add(an); yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 65))) # Michael S. Branicky, Nov 07 2022

A320864 Powers of 2 with initial digit 8.

Original entry on oeis.org

8, 8192, 8388608, 8589934592, 8796093022208, 81129638414606681695789005144064, 83076749736557242056487941267521536, 85070591730234615865843651857942052864, 87112285931760246646623899502532662132736, 89202980794122492566142873090593446023921664

Offset: 1

Muniru A Asiru, Nov 21 2018

Muniru A Asiru, Table of n, a(n) for n = 1..170
Index to divisibility sequences

Cf. A000079 (powers of 2), A008952 (leading digit of 2^n), A217401 (numbers starting with 8).

Powers of 2 with initial digit k, (k = 1..8): A067488, A067480, A320859, A320860, A320861, A320862, A320863, this sequence.

Filtered(List([0..200],n->2^n),i->ListOfDigits(i)[1]=8);

select(x->"8"=""||x[1],[2^n$n=0..200])[];

Select[2^Range[200], IntegerDigits[#][[1]] == 8 &] (* Amiram Eldar, Nov 21 2018 *)

select(x->(digits(x)[1]==8), vector(200, n, 2^n)) \\ Michel Marcus, Nov 21 2018

A138028 The array of the most significant digit of n^k read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Feb 10 2008

			n\k
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... .
1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, ... .
1, 3, 9, 2, 8, 2, 7, 2, 6, 1, 5, 1, 5, 1, 4, 1, 4, 1, 3, 1, 3, ... .
1, 4, 1, 6, 2, 1, 4, 1, 6, 2, 1, 4, 1, 6, 2, 1, 4, 1, 6, 2, 1, ... .
1, 5, 2, 1, 6, 3, 1, 7, 3, 1, 9, 4, 2, 1, 6, 3, 1, 7, 3, 1, 9, ... .
1, 6, 3, 2, 1, 7, 4, 2, 1, 1, 6, 3, 2, 1, 7, 4, 2, 1, 1, 6, 3, ... .
1, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1, ... .
1, 9, 8, 7, 6, 5, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, ... .
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... .
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, ... .
1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 6, 7, 8, 1, 1, 1, 1, 2, 2, 3, 3, ... .
......................................................................

Cf. A008952, A060956, A138029, A061505.

f[n_, k_] := Quotient[n^k, 10^Floor[k*Log[10, n]]]; Table[f[ n - k, k], {n, 14}, {k, 0, n - 1}] // Flatten

A320865 Powers of 2 with initial digit 9.

Original entry on oeis.org

9007199254740992, 9223372036854775808, 9444732965739290427392, 9671406556917033397649408, 9903520314283042199192993792, 91343852333181432387730302044767688728495783936, 93536104789177786765035829293842113257979682750464

Offset: 1

Muniru A Asiru, Nov 21 2018

Muniru A Asiru, Table of n, a(n) for n = 1..150
Index to divisibility sequences

Cf. A000079 (powers of 2), A008952 (leading digit of 2^n), A217402 (numbers starting with 9).

Powers of 2 with initial digit k, (k = 1..9): A067488, A067480, A320859, A320860, A320861, A320862, A320863, A320864, this sequence.

Filtered(List([0..200],n->2^n),i->ListOfDigits(i)[1]=9);

select(x->"9"=""||x[1],[2^n$n=0..200])[];

Select[2^Range[200], IntegerDigits[#][[1]] == 9 &] (* Amiram Eldar, Nov 21 2018 *)

select(x->(digits(x)[1]==9), vector(200, n, 2^n)) \\ Michel Marcus, Nov 21 2018

A354782 Second digit from left in decimal expansion of 2^n (n >= 4).

Original entry on oeis.org

6, 2, 4, 2, 5, 1, 0, 0, 0, 1, 6, 2, 5, 3, 6, 2, 0, 0, 1, 3, 6, 3, 7, 3, 6, 3, 0, 1, 2, 5, 7, 4, 8, 3, 7, 4, 0, 1, 3, 7, 7, 5, 0, 4, 8, 6, 1, 2, 5, 0, 8, 6, 2, 4, 8, 7, 1, 3, 6, 2, 8, 6, 3, 4, 9, 9, 1, 3, 7, 4, 8, 7, 5, 5, 0, 0, 2, 4, 8, 6, 9, 8, 7, 5, 0, 1, 2, 4, 9, 9, 9, 9, 9, 5, 1, 3, 2, 5, 0, 0, 0, 0, 1, 6, 2, 4, 2, 5, 1

Offset: 4

N. J. A. Sloane, Jul 07 2022, following a suggestion from Alexander Wajnberg

			2^4 = 16, so a(4) = 6. 2^5 = 32, so a(5) = 2.

Alois P. Heinz, Table of n, a(n) for n = 4..10000

Cf. A000079, A000689, A008952, A034887, A160590.

a:= n-> parse(""||(2^n)[2]):
seq(a(n), n=4..112);  # Alois P. Heinz, Jul 07 2022

A354782[n_]:=IntegerDigits[2^n][[2]];Array[A354782,100,4] (* Paolo Xausa, Oct 22 2023 *)

def A354782(n): return int(str(1<Chai Wah Wu, Jul 07 2022

cp's OEIS Frontend

A320863 Powers of 2 with initial digit 7.

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A362871 Leading digit of 6^n.

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A363093 Leading digit of 7^n.

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A367361 Comma transform of powers of 2.

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A141053 Most-significant decimal digit of Fibonacci(5n+3).

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A247243 a(n) = smallest positive integer k not already in the sequence such that the decimal expansion of 2^(n-1) begins with k.

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A320864 Powers of 2 with initial digit 8.

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A138028 The array of the most significant digit of n^k read by antidiagonals.

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