A074118 -id:A074118 - 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)

A074116 Largest n-digit power of 2.

Original entry on oeis.org

8, 64, 512, 8192, 65536, 524288, 8388608, 67108864, 536870912, 8589934592, 68719476736, 549755813888, 8796093022208, 70368744177664, 562949953421312, 9007199254740992, 72057594037927936, 576460752303423488, 9223372036854775808, 73786976294838206464, 590295810358705651712

Offset: 1

Amarnath Murthy, Aug 27 2002

The exponents are given in A066343. - Evgeny Kapun, Jan 16 2017

An equivalent definition (which was formerly the definition of A074113): "Smallest n-digit number of the form p^a*q^b... with the maximum value of a+b+.... where p, q etc. are primes. If a,b,c,... are the indices in the signature prime factorization then a+b+c ... is a maximum." That this is the same sequence follows from the inequality p^a*q^b... >= 2^(a+b+...) and the fact that there always exists a power of 2 between two consecutive powers of 10.

Evgeny Kapun, Table of n, a(n) for n = 1..1000

Cf. A066343, A067488, A074113, A074114, A074117, A074118.

Last[#]&/@(With[{l2=2^Range[80]},Table[Select[l2,IntegerLength[#] == n&], {n,22}]]) (* Harvey P. Dale, Jul 17 2011 *)

a(n) = 2^A066343(n).

Edited by R. J. Mathar, Feb 13 2008, Max Alekseyev, Mar 10 2009, Harvey P. Dale, Jul 17 2011, Evgeny Kapun, Jan 16 2017, and N. J. A. Sloane, Jan 18 2017

A074117 Smallest n-digit power of 3.

Original entry on oeis.org

1, 27, 243, 2187, 19683, 177147, 1594323, 14348907, 129140163, 1162261467, 10460353203, 282429536481, 2541865828329, 22876792454961, 205891132094649, 1853020188851841, 16677181699666569, 150094635296999121, 1350851717672992089, 12157665459056928801

Offset: 1

Amarnath Murthy, Aug 27 2002

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A067488, A074116, A074118.

a(n) = 3^ceiling((n-1)*log(10)/log(3)). - Andrew Howroyd, Jan 16 2020

a(1) corrected and terms a(16) and beyond from Andrew Howroyd, Jan 16 2020

A054965 Beatty sequence for log_3(10), i.e., for 1/log_10(3); so largest exponent of 3 which produces an n-digit decimal number.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 132, 134, 136

Offset: 1

Henry Bottomley, Dec 13 2002

			log_10(3) = 0.477121... so a(11) = floor(11/0.477121...) = floor(23.0549...) = 23; 3^23 = 94143178827 is the largest 11 decimal digit power of 3.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences.

Cf. A062153, A066343, A074118, A152566.

Floor[Range[100]*Log[3, 10]] (* Paolo Xausa, Jul 11 2024 *)

a(n) = n*log(10)\log(3); \\ Michel Marcus, Aug 03 2017

a(n) = floor(n/log_10(3)) = log_3(A074118(n)) = A062153(A074118(n)).

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A067488 Powers of 2 with initial digit 1.

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A074116 Largest n-digit power of 2.

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A074117 Smallest n-digit power of 3.

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A054965 Beatty sequence for log_3(10), i.e., for 1/log_10(3); so largest exponent of 3 which produces an n-digit decimal number.

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