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

Author

Amarnath Murthy, Feb 09 2002

Keywords

• base
• easy
• nonn

Comments

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

Links

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

Crossrefs

Cf. A000079, A067497, A055642,

Other initial digits: A067480, A067481, A067482, A067483, A067484, A067485, A067486, A067487, A074116.

Cf. A074117, A074118.

Programs

GAP

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

Magma

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

Mathematica

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

PARI

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

Scala

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

Formula

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)