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%I #34 Feb 18 2021 00:20:32
%S 0,1,81,625,7776,16807,46656,59049,1679616,1475789056,6975757441,
%T 137858491849,576650390625,41426511213649,2384185791015625,
%U 150094635296999121,10260628712958602189,32768000000000000000,243569224216081305397,655360000000000000000
%N Base-2 Reacher numbers: numbers that are powers of the sum of their base-2 digits.
%C Named for fictional character Jack Reacher in the series of novels by Lee Child.
%C There are 2709 terms with 10,000 or fewer digits; a(2709) = 15402^2388. - _Charles R Greathouse IV_, Nov 26 2016
%D Lee Child, Bad Luck and Trouble, Delacorte Press, 2007. In this book, the main character, Jack Reacher, likes the number 81 because it is the square of the sum of its base-10 digits.
%H Hiroaki Yamanouchi and Charles R Greathouse IV, <a href="/A256590/b256590.txt">Table of n, a(n) for n = 1..388</a> (first 141 terms from Hiroaki Yamanouchi)
%H J. Shallit, <a href="http://recursed.blogspot.ca/2007/09/mathematics-in-jack-reacher-novel.html">Mathematics in a Jack Reacher Novel</a>, blog post, Sep 08 2007.
%t fQ[n_] := Block[{wt = DigitCount[n, 2, 1]},
%t Which[n <= 1, True, wt <= 1, False, True, IntegerQ@ Log[wt, n]]]; Select[Range[10^5], fQ] (* _Michael De Vlieger_, Apr 04 2015 *)
%o (PARI) is(n)= n<=1 || (ispower(n,,&r) && (r==hammingweight(n) || (r^ispower(n=hammingweight(n))==n && n>1))) \\ _Michel Marcus_ and _M. F. Hasler_, Apr 04 2015
%o (PARI) list(lim)=my(v=List([0,1]),H,t); for(e=3,logint(lim\=1,3), for(b=2,min(solve(x=e,lim,x-e*log(x)/log(2)-1),sqrtnint(lim,e)), H=hammingweight(t=b^e); if(H>1 && b^valuation(H,b)==H, listput(v,t)))); Set(v) \\ _Charles R Greathouse IV_, Nov 26 2016
%Y Cf. A023106.
%K nonn,base
%O 1,3
%A _Jeffrey Shallit_, Apr 03 2015
%E a(1) prepended and a(14)-a(20) added by _Hiroaki Yamanouchi_, Apr 03 2015
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