A049445 Numbers n with property that the number of 1's in binary expansion of n (see A000120) divides n.

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

Offset

1,2

Comments

If instead of base 2 we take base 10, then we have the so-called 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 and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quarterly, Vol. 56, No. 2 (2018), pp. 163-166.

Jean-Marie De Koninck, Nicolas Doyon and Imre Kátai, On the counting function for the Niven numbers, Acta Arithmetica, Vol. 106, No. 3 (2003), pp. 265-275.

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 d-bit 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

De Koninck et al. (2003) proved that the number of base-b Niven numbers not exceeding x, N_b(x), is asymptotically equal to ((2*log(b)/(b-1)^2) * Sum_{j=1..b-1} gcd(j, b-1) + o(1)) * x/log(x). For b=2, N_2(n) ~ (2*log(2) + o(1)) * x/log(x). Therefore, the constant c mentioned above is indeed 1/(2*log(2)). - Amiram Eldar, Aug 16 2020

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] (* Jean-François 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 !! (n-1)
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 * A356448 A340521 A002174

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