

A049445


Numbers n with property that the number of 1's in binary expansion of n (see A000120) divides n.


33



1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 21, 24, 32, 34, 36, 40, 42, 48, 55, 60, 64, 66, 68, 69, 72, 80, 81, 84, 92, 96, 108, 110, 115, 116, 120, 126, 128, 130, 132, 136, 138, 144, 155, 156, 160, 162, 168, 172, 180, 184, 185, 192, 204, 205, 212, 216, 220, 222, 228
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OFFSET

1,2


COMMENTS

If instead of base 2 we take base 10, then we have the socalled Harshad or Niven numbers (i.e., positive integers divisible by the sum of their digits; A005349).  Emeric Deutsch, Apr 11 2007
A199238(a(n)) = 0.  Reinhard Zumkeller, Nov 04 2011


LINKS

Indranil Ghosh, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
Paul Dalenberg, Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quart. (2018) Vol. 56, No. 2, 163166.


FORMULA

{n: A000120(n)  n}.  R. J. Mathar, Mar 03 2008
a(n) seems to be asymptotic to c*n*log(n) where 0.7 < c < 0.8.  Benoit Cloitre, Jan 22 2003
Heuristically, c should be 1/(2*log(2)), since a random dbit number should have probability approximately 2/d of being in the sequence.  Robert Israel, Aug 22 2014
A049445 = { n: A199238(n)=0 }.  M. F. Hasler, Oct 09 2012


EXAMPLE

20 is in the sequence because 20 is written 10100 in binary and 1 + 1 = 2, which divides 20.
21 is in the sequence because 21 is written 10101 in binary and 1 + 1 + 1 = 3, which divides 21.
22 is not in the sequence because 22 is written 10110 in binary 1 + 1 + 1 = 3, which does not divide 22.


MAPLE

a:=proc(n) local n2: n2:=convert(n, base, 2): if n mod add(n2[i], i=1..nops(n2)) = 0 then n else fi end: seq(a(n), n=1..300); # Emeric Deutsch, Apr 11 2007


MATHEMATICA

binHarshadQ[n_] := Divisible[n, Count[IntegerDigits[n, 2], 1]]; Select[Range[228], binHarshadQ] (* JeanFrançois Alcover, Dec 01 2011 *)
Select[Range[300], Divisible[#, DigitCount[#, 2, 1]]&] (* Harvey P. Dale, Mar 20 2016 *)


PROG

(PARI) for(n=1, 1000, b=binary(n); l=length(b); if(n%sum(i=1, l, component(b, i))==0, print1(n, ", ")))
(PARI) is_A049445(n)={n%norml2(binary(n))==0} \\ M. F. Hasler, Oct 09 2012
(PARI) isok(n) = ! (n % hammingweight(n)); \\ Michel Marcus, Feb 10 2016
(Haskell)
a049445 n = a049445_list !! (n1)
a049445_list = map (+ 1) $ elemIndices 0 a199238_list
 Reinhard Zumkeller, Nov 04 2011
(Python)
A049445 = [n for n in range(1, 10**5) if not n % sum([int(d) for d in bin(n)[2:]])] # Chai Wah Wu, Aug 22 2014


CROSSREFS

Cf. A000120, A005349, A199238.
Sequence in context: A186384 A011860 A259278 * A002174 A002202 A049225
Adjacent sequences: A049442 A049443 A049444 * A049446 A049447 A049448


KEYWORD

nonn,easy,nice,base


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Michael Somos
Edited by N. J. A. Sloane, Oct 07 2005 and May 16 2008


STATUS

approved



