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

0, 1, 81, 625, 7776, 16807, 46656, 59049, 1679616, 1475789056, 6975757441, 137858491849, 576650390625, 41426511213649, 2384185791015625, 150094635296999121, 10260628712958602189, 32768000000000000000, 243569224216081305397, 655360000000000000000

Offset: 1

Author

Jeffrey Shallit, Apr 03 2015

Keywords

• nonn
• base

Comments

Named for fictional character Jack Reacher in the series of novels by Lee Child.

There are 2709 terms with 10,000 or fewer digits; a(2709) = 15402^2388. - Charles R Greathouse IV, Nov 26 2016

References

• 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.

Links

• Hiroaki Yamanouchi and Charles R Greathouse IV, Table of n, a(n) for n = 1..388 (first 141 terms from Hiroaki Yamanouchi)
• J. Shallit, Mathematics in a Jack Reacher Novel, blog post, Sep 08 2007.

Crossrefs

Cf. A023106, A161792.

Programs

Mathematica

fQ[n_] := Block[{wt = DigitCount[n, 2, 1]},
Which[n <= 1, True, wt <= 1, False, True, IntegerQ@ Log[wt, n]]]; Select[Range[10^5], fQ] (* Michael De Vlieger, Apr 04 2015 *)

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

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

Extensions

a(1) prepended and a(14)-a(20) added by Hiroaki Yamanouchi, Apr 03 2015