A067482 -id:A067482 - 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

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

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

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

A067490 Powers of 4 with initial digit 1.

Original entry on oeis.org

1, 16, 1024, 16384, 1048576, 16777216, 1073741824, 17179869184, 1099511627776, 17592186044416, 1125899906842624, 18014398509481984, 1152921504606846976, 18446744073709551616, 1180591620717411303424, 18889465931478580854784, 1208925819614629174706176

Offset: 1

Amarnath Murthy, Feb 09 2002

Muniru A Asiru, Table of n, a(n) for n = 1..400
Index to divisibility sequences

Cf. A067480, A067481, A067482, A067483, A067484, A067485, A067486, A067487, A067488, A067489.

Filtered(List([0..40],n->4^n),i->ListOfDigits(i)[1]=1); # Muniru A Asiru, Oct 22 2018

select(x-> "1"=""||x[1],[4^n$n=0..60])[];  # Alois P. Heinz, Oct 22 2018

a(n+1)/a(n) is in {16, 64, 1024}, so 16^n <= a(n+1) < 1024^n. Asymptotically, the exponent should be 100; I can prove that 99^n << a(n) << 101^n. [Charles R Greathouse IV, Jan 19 2012]

a(16) inserted by Muniru A Asiru, Oct 22 2018

A067491 Powers of 5 with initial digit 1.

Original entry on oeis.org

1, 125, 15625, 1953125, 1220703125, 152587890625, 19073486328125, 11920928955078125, 1490116119384765625, 186264514923095703125, 116415321826934814453125, 14551915228366851806640625, 1818989403545856475830078125, 1136868377216160297393798828125

Offset: 1

Amarnath Murthy, Feb 09 2002

Muniru A Asiru, Table of n, a(n) for n = 1..300

Cf. A067480, A067481, A067482, A067483, A067484, A067485, A067486, A067487, A067488, A067489, A067490.

Filtered(List([0..40],n->5^n),i->ListOfDigits(i)[1]=1); # Muniru A Asiru, Oct 21 2018

select(x-> "1"=""||x[1], [5^n$n=0..50])[];  # Alois P. Heinz, Oct 21 2018

Offset 1 from Michel Marcus, Oct 19 2018

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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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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A067489 Powers of 3 with initial digit 1.

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