A067480 -id:A067480 - cp's OEIS frontend

A067488 Powers of 2 with initial digit 1.

1, 16, 128, 1024, 16384, 131072, 1048576, 16777216, 134217728, 1073741824, 17179869184, 137438953472, 1099511627776, 17592186044416, 140737488355328, 1125899906842624, 18014398509481984, 144115188075855872, 1152921504606846976, 18446744073709551616

Offset: 1

Amarnath Murthy, Feb 09 2002

Also smallest n-digit power of 2.

For each range 10^(n-1) to 10^n-1 there exists exactly 1 power of 2 with first digit 1 (floor(log_10(a(n))) = n-1). As such, the density of this sequence relative to all powers of 2 (A000079) is log(2)/log(10) (0.301..., A007524), which is prototypical of Benford's Law. - Charles L. Hohn, Jul 23 2024

Muniru A Asiru, Table of n, a(n) for n = 1..993
Index to divisibility sequences

Cf. A000079, A067497, A055642,
Other initial digits: A067480, A067481, A067482, A067483, A067484, A067485, A067486, A067487, A074116.
Cf. A074117, A074118.

Filtered(List([0..60],n->2^n),i->ListOfDigits(i)[1]=1); # Muniru A Asiru, Oct 22 2018

[2^n: n in [0..100] | Intseq(2^n)[#Intseq(2^n)] eq 1]; // Vincenzo Librandi, Dec 31 2024

Select[2^Range[0, 70], First[IntegerDigits[#]] == 1 &]  (* Harvey P. Dale, Mar 14 2011 *)

a(n)=2^ceil((n-1)*log(10)/log(2)) \\ Charles R Greathouse IV, Apr 08 2012

(List.fill(50)(2: BigInt)).scanLeft(1: BigInt)( * ).filter(.toString.startsWith("1")) // _Alonso del Arte, Jan 16 2020

a(n) = 2^ceiling((n-1)*log(10)/log(2)). - Benoit Cloitre, Aug 29 2002
From Charles L. Hohn, Jun 09 2024: (Start)
a(n) = 2^A067497(n-1).
A055642(a(n)) = n. (End)

A067483 Powers of 5 with initial digit 5.

Original entry on oeis.org

5, 59604644775390625, 582076609134674072265625, 5684341886080801486968994140625, 55511151231257827021181583404541015625, 542101086242752217003726400434970855712890625

Offset: 1

Amarnath Murthy, Feb 09 2002

Each term also has final digit 5. - Muniru A Asiru, Oct 13 2018

Muniru A Asiru, Table of n, a(n) for n = 1..114

Subsequence of A000351 (powers of 5).

Similar entries with another digit: A067480 (2), A067481 (3), A067482 (4).

k:=5;; Filtered(List([0..100],n->k^n),i->ListOfDigits(i)[1]=k); # Muniru A Asiru, Oct 06 2018

Select[5^Range[70],First[IntegerDigits[#]]==5&]  (* Harvey P. Dale, Apr 01 2011 *)

lista(nn) = {for (n=1, nn, if (digits(p=5^n)[1] == 5, print1(p, ", ")););} \\ Michel Marcus, Oct 14 2018

Edited by Frank Ellermann, Feb 11 2002

One more term from Harvey P. Dale, Apr 01 2011

A067497 Smallest k for which 2^k is n+1 decimal digits long, and equivalently numbers k such that 1 is the first digit of 2^k.

Original entry on oeis.org

0, 4, 7, 10, 14, 17, 20, 24, 27, 30, 34, 37, 40, 44, 47, 50, 54, 57, 60, 64, 67, 70, 74, 77, 80, 84, 87, 90, 94, 97, 100, 103, 107, 110, 113, 117, 120, 123, 127, 130, 133, 137, 140, 143, 147, 150, 153, 157, 160, 163, 167, 170, 173, 177, 180, 183, 187, 190, 193, 196

Offset: 0

Benoit Cloitre, Feb 22 2002

The asymptotic density of this sequence is log_10(2) = 0.301029... (A007524). - Amiram Eldar, Jan 27 2021

Muniru A Asiru, Table of n, a(n) for n = 0..5000

Cf. A000079, A007524, A066343, A067480, A067488, A129344.

Filtered([0..200],n->ListOfDigits(2^n)[1]=1); # Muniru A Asiru, Oct 22 2018

a[n_] := Block[{k = 0}, While[ Floor[Log[10, 2^k] + 1] < n, k++ ]; k]; Table[ a[n], {n, 1, 61}]
Table[Ceiling[n*Log[2, 10]], {n, 0, 59}] (* Jean-François Alcover, Jan 29 2014, after Vladeta Jovovic *)

for(n=0,500, if(floor(2^n/10^(floor(n*log(2)/log(10))))==1,print1(n,", ")))

a(n) = ceil(n*log(10)/log(2)); \\ Michel Marcus, May 13 2017

def A067497(n): return (10**n-1).bit_length() # Chai Wah Wu, Apr 02 2023

[ceil(n*log(10)/log(2)) for n in range(0, 60)] # Stefano Spezia, Aug 31 2024

a(n) = ceiling(n*log_2(10)). - Vladeta Jovovic, Jun 20 2002

a(n) = log_2(A067488(n+1)). - Charles L. Hohn, Jun 09 2024

Additional comments from Lekraj Beedassy, Jun 20 2002 and from Rick Shephard, Jun 27 2002

A067484 Powers of 6 with initial digit 6.

Original entry on oeis.org

6, 60466176, 609359740010496, 6140942214464815497216, 61886548790943213277031694336, 623673825204293256669089197883129856, 6285195213566005335561053533150026217291776, 63340286662973277706162286946811886609896461828096

Offset: 1

Amarnath Murthy, Feb 09 2002

The geometric progression formula a(n)=10077696*a(n-1) does NOT hold if n=20, 40, 59, 79, 98, etc. - R. J. Mathar, Jun 24 2009

Muniru A Asiru, Table of n, a(n) for n = 1..89
Tanya Khovanova, Non Recursions

Cf. A067480, A067481, A067482, A067483.

k:=6;; Filtered(List([0..80],n->k^n),i->ListOfDigits(i)[1]=k); # Muniru A Asiru, Oct 18 2018

A067484 := proc(n) local p6,p,a ; if n = 1 then 6; else p6 := procname(n-1) ; ifactors(p6)[2] ; p := op(2,op(1,%)) ; for a from p+1 do p6 := 6^a ; convert(%,base,10) ; if op(-1,%) = 6 then RETURN(p6) ; fi; od: fi; end: # R. J. Mathar, Jun 24 2009

Select[6^Range[100],First[IntegerDigits[#]]==6&] (* Harvey P. Dale, Aug 14 2018 *)

More terms from Benoit Cloitre, Feb 22 2002
a(8) from Harvey P. Dale, Aug 14 2018
Offset changed to 1 by Muniru A Asiru, Oct 19 2018

A067485 Powers of 7 with initial digit 7.

Original entry on oeis.org

7, 79792266297612001, 7730993719707444524137094407, 749048330965186233494494102694564493649, 72574551534231909331741171093173785967490646405143

Offset: 1

Amarnath Murthy, Feb 09 2002

Muniru A Asiru, Table of n, a(n) for n = 1..70

Cf. A067480, A067481, A067482, A067483, A067484.

Filtered(List([0..100],n->7^n),i->ListOfDigits(i)[1]=7); # Muniru A Asiru, Oct 22 2018

Select[7^Range[80],First[IntegerDigits[#]]==7&] (* Harvey P. Dale, Apr 16 2013 *)

More terms from Benoit Cloitre, Feb 22 2002

A067486 Powers of 8 with initial digit 8.

Original entry on oeis.org

8, 8589934592, 85070591730234615865843651857942052864, 842498333348457493583344221469363458551160763204392890034487820288, 8343699359066055009355553539724812947666814540455674882605631280555545803830627148527195652096

Offset: 1

Amarnath Murthy, Feb 09 2002

Muniru A Asiru, Table of n, a(n) for n = 1..57

Cf. A067480, A067481, A067482, A067483, A067484, A067485.

Filtered(List([0..110],n->8^n),i->ListOfDigits(i)[1]=8); # Muniru A Asiru, Oct 22 2018

Select[8^Range[150],First[IntegerDigits[#]]==8&]  (* Harvey P. Dale, Dec 26 2010 *)

for(n=1,200, if(floor((8^n)/10^floor(log((8^n))/log(10)))==8,print1(8^n,",")))

More terms from Benoit Cloitre, Feb 28 2002

A067487 Powers of 9 with initial digit 9.

Original entry on oeis.org

9, 984770902183611232881, 969773729787523602876821942164080815560161, 955004950796825236893190701774414011919935138974343129836853841, 940461086986004843694934910131056317906479029659199959555574885740211572136210345921

Offset: 1

Amarnath Murthy, Feb 09 2002

Muniru A Asiru, Table of n, a(n) for n = 1..48

Cf. A067480, A067481, A067482, A067483, A067484, A067485, A067486.

Filtered(List([0..100],n->9^n),i->ListOfDigits(i)[1]=9); # Muniru A Asiru, Oct 21 2018

[9^n: n in [1..100] | Intseq(9^n)[#Intseq(9^n)] eq 9]; // Vincenzo Librandi, Oct 22 2018

 Select[9^Range[100], First[IntegerDigits[#]]==9 &] (* Vincenzo Librandi, Oct 22 2018 *)

More terms from Benoit Cloitre, Feb 28 2002

A067469 Numbers k such that 2 is the first digit of 2^k.

Original entry on oeis.org

1, 8, 11, 18, 21, 28, 31, 38, 41, 48, 51, 58, 61, 68, 71, 81, 91, 101, 104, 111, 114, 121, 124, 131, 134, 141, 144, 151, 154, 161, 164, 171, 174, 184, 194, 197, 204, 207, 214, 217, 224, 227, 234, 237, 244, 247, 254, 257, 264, 267, 277, 287, 297, 300, 307, 310

Offset: 1

Benoit Cloitre, Feb 22 2002

The asymptotic density of this sequence is log_10(3/2) = 0.176091... (A154580 - 1). - Amiram Eldar, Jan 27 2021

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Cf. A067480, A154580.

Filtered([1..310],n->ListOfDigits(2^n)[1]=2); # Muniru A Asiru, Oct 14 2018

Select[Range@ 310, First@ IntegerDigits[2^#] == 2 &] (* Michael De Vlieger, Oct 15 2018 *)

is(n)=digits(2^n)[1]==2 \\ Charles R Greathouse IV, Jul 29 2013

A067489 Powers of 3 with initial digit 1.

Original entry on oeis.org

1, 19683, 177147, 1594323, 14348907, 129140163, 1162261467, 10460353203, 1853020188851841, 16677181699666569, 150094635296999121, 1350851717672992089, 12157665459056928801, 109418989131512359209

Offset: 1

Amarnath Murthy, Feb 09 2002

Muniru A Asiru, Table of n, a(n) for n = 1..625

Cf. A067480, A067481, A067482, A067483, A067484, A067485, A067486, A067487, A067488.

Filtered(List([0..50],n->3^n),i->ListOfDigits(i)[1]=1); # Muniru A Asiru, Oct 22 2018

Select[3^Range[0,5*10^6],First[IntegerDigits[#]]==1&] (* Harvey P. Dale, Oct 09 2015 *)

Offset 1 from Michel Marcus, Oct 19 2018

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).

cp's OEIS Frontend

A067488 Powers of 2 with initial digit 1.

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A067483 Powers of 5 with initial digit 5.

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A067497 Smallest k for which 2^k is n+1 decimal digits long, and equivalently numbers k such that 1 is the first digit of 2^k.

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A067484 Powers of 6 with initial digit 6.

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A067485 Powers of 7 with initial digit 7.

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A067486 Powers of 8 with initial digit 8.

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A067487 Powers of 9 with initial digit 9.

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