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

1,3

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.

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 *)

Prog

(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

Crossrefs

Cf. A023106.

Sequence in context: A359607 A016756 A182647 * A322240 A185856 A185848

Adjacent sequences: A256587 A256588 A256589 * A256591 A256592 A256593

Keyword

nonn,base

Author

Jeffrey Shallit, Apr 03 2015

Extensions

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

Status

approved