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A256590 Base-2 Reacher numbers: numbers that are powers of the sum of their base-2 digits.

%I A256590 #37 Jul 17 2025 14:47:50
%S A256590 0,1,81,625,7776,16807,46656,59049,1679616,1475789056,6975757441,
%T A256590 137858491849,576650390625,41426511213649,2384185791015625,
%U A256590 150094635296999121,10260628712958602189,32768000000000000000,243569224216081305397,655360000000000000000
%N A256590 Base-2 Reacher numbers: numbers that are powers of the sum of their base-2 digits.
%C A256590 Named for fictional character Jack Reacher in the series of novels by Lee Child.
%C A256590 There are 2709 terms with 10,000 or fewer digits; a(2709) = 15402^2388. - _Charles R Greathouse IV_, Nov 26 2016
%D A256590 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 A256590 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 A256590 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 A256590 fQ[n_] := Block[{wt = DigitCount[n, 2, 1]},
%t A256590 Which[n <= 1, True, wt <= 1, False, True, IntegerQ@ Log[wt, n]]]; Select[Range[10^5], fQ] (* _Michael De Vlieger_, Apr 04 2015 *)
%o A256590 (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 A256590 (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 A256590 Cf. A023106, A161792.
%K A256590 nonn,base
%O A256590 1,3
%A A256590 _Jeffrey Shallit_, Apr 03 2015
%E A256590 a(1) prepended and a(14)-a(20) added by _Hiroaki Yamanouchi_, Apr 03 2015