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%I #50 Jan 03 2025 02:18:00
%S 1,16,128,1024,16384,131072,1048576,16777216,134217728,1073741824,
%T 17179869184,137438953472,1099511627776,17592186044416,
%U 140737488355328,1125899906842624,18014398509481984,144115188075855872,1152921504606846976,18446744073709551616
%N Powers of 2 with initial digit 1.
%C Also smallest n-digit power of 2.
%C 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
%H Muniru A Asiru, <a href="/A067488/b067488.txt">Table of n, a(n) for n = 1..993</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F a(n) = 2^ceiling((n-1)*log(10)/log(2)). - _Benoit Cloitre_, Aug 29 2002
%F From _Charles L. Hohn_, Jun 09 2024: (Start)
%F a(n) = 2^A067497(n-1).
%F A055642(a(n)) = n. (End)
%t Select[2^Range[0, 70], First[IntegerDigits[#]] == 1 &] (* _Harvey P. Dale_, Mar 14 2011 *)
%o (PARI) a(n)=2^ceil((n-1)*log(10)/log(2)) \\ _Charles R Greathouse IV_, Apr 08 2012
%o (GAP) Filtered(List([0..60],n->2^n),i->ListOfDigits(i)[1]=1); # _Muniru A Asiru_, Oct 22 2018
%o (Scala) (List.fill(50)(2: BigInt)).scanLeft(1: BigInt)(_ * _).filter(_.toString.startsWith("1")) // _Alonso del Arte_, Jan 16 2020
%o (Magma) [2^n: n in [0..100] | Intseq(2^n)[#Intseq(2^n)] eq 1]; // _Vincenzo Librandi_, Dec 31 2024
%Y Cf. A000079, A067497, A055642,
%Y Other initial digits: A067480, A067481, A067482, A067483, A067484, A067485, A067486, A067487, A074116.
%Y Cf. A074117, A074118.
%K base,easy,nonn,changed
%O 1,2
%A _Amarnath Murthy_, Feb 09 2002