A067483 -id:A067483 - cp's OEIS frontend

A067480 Powers of 2 with initial digit 2.

2, 256, 2048, 262144, 2097152, 268435456, 2147483648, 274877906944, 2199023255552, 281474976710656, 2251799813685248, 288230376151711744, 2305843009213693952, 295147905179352825856, 2361183241434822606848, 2417851639229258349412352, 2475880078570760549798248448

Offset: 1

Amarnath Murthy, Feb 09 2002

Muniru A Asiru, Table of n, a(n) for n = 1..400

Cf. A000079, A067469, A067483.

k:=2;; Filtered(List([0..100],n->k^n),i->ListOfDigits(i)[1]=k); # Muniru A Asiru, Oct 19 2018

[2^n: n in [1..100] | Intseq(2^n)[#Intseq(2^n)] eq 2]; // Vincenzo Librandi, Oct 22 2018

Select[Table[2^n,{n,75}],First@IntegerDigits[#]==2 &] (* Jayanta Basu, May 19 2013 *)
Select[2^Range[100], First[IntegerDigits[#]]==2 &] (* Vincenzo Librandi, Oct 22 2018 *)

lista(nn) = {for (n=0, nn, if (digits(x=2^n)[1] == 2, print1(x, ", ")););} \\ Michel Marcus, Oct 20 2018

A067488 Powers of 2 with initial digit 1.

Original entry on oeis.org

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)

A067482 Powers of 4 with initial digit 4.

Original entry on oeis.org

4, 4096, 4194304, 4294967296, 4398046511104, 4503599627370496, 4611686018427387904, 4722366482869645213696, 4835703278458516698824704, 4951760157141521099596496896, 401734511064747568885490523085290650630550748445698208825344

Offset: 1

Amarnath Murthy, Feb 09 2002

Appearances to the contrary, this sequence is not recursive with a geometric progression factor of 1024. Examples of a(n+1)/a(n) which are not 1024 are a(10)/a(9) = a(20)/a(19) = a(30)/a(29) = a(57)/a(56) = a(67)/a(66) = 2^106 or a(39)/a(38) = a(48)/a(47)= a(86)/a(85) = 2^116. - R. J. Mathar, Jan 10 2008

Therefore this sequence differs from A013830. - Georg Fischer, Oct 06 2018

Muniru A Asiru, Table of n, a(n) for n = 1..71
Tanya Khovanova, Non Recursions

Cf. A000302, A067483, A013830, A242594.

Filtered(List([0..100],n->4^n),i->ListOfDigits(i)[1]=4); # Muniru A Asiru, Oct 06 2018

DeleteCases[Table[If[StringStartsQ[ToString[4^n],"4"],4^n], {n, 100}], Null] (* Stefano Spezia, Oct 06 2018 *)

a(n) = 4^A242594(n).

A067481 Powers of 3 with initial digit 3.

Original entry on oeis.org

3, 387420489, 3486784401, 31381059609, 36472996377170786403, 328256967394537077627, 381520424476945831628649898809, 3433683820292512484657849089281, 30903154382632612361920641803529, 3990838394187339929534246675572349035227, 35917545547686059365808220080151141317043

Offset: 1

Amarnath Murthy, Feb 09 2002

Alois P. Heinz, Table of n, a(n) for n = 1..261

Cf. A000244, A067483.

Cf. A054565.

[3^n: n in [1..100] | Intseq(3^n)[#Intseq(3^n)] eq 3]; // Vincenzo Librandi, Oct 22 2018

Select[3^Range[70], First[IntegerDigits[#]]==3 &] (* Vincenzo Librandi, Oct 22 2018 *)

lista(nn) = {for (n=0, nn, if (digits(x=3^n)[1] == 3, print1(x, ", ")););} \\ Michel Marcus, Oct 22 2018

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

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

cp's OEIS Frontend

A067480 Powers of 2 with initial digit 2.

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A067488 Powers of 2 with initial digit 1.

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A067482 Powers of 4 with initial digit 4.

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A067481 Powers of 3 with initial digit 3.

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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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