A089042 Composite numbers such that all divisors >1 have the same number of 1's in binary representation.

4, 8, 9, 16, 32, 49, 64, 128, 133, 256, 259, 512, 961, 1024, 2048, 2059, 2449, 3713, 4096, 4681, 4867, 6169, 6241, 8192, 8401, 8773, 9353, 10261, 10561, 12307, 12449, 16129, 16384, 16459, 16531, 16771, 18467, 20491, 24649, 24721, 24961, 25217

Offset

1,1

Comments

A000120(d)=constant for all d with 1<d<=a(n) and d|a(n).

Are there terms with more than 2 distinct prime factors?

No terms with omega(n)>2 up to 10000000. - Michel Marcus, Jun 05 2013

From Robert Israel, Dec 01 2015: (Start)

The only term divisible by 3 is 9.

The terms divisible by 2 are 2^k for k > 1.

There are no terms divisible by 5. (End)

Links

Harvey P. Dale and Robert Israel, Table of n, a(n) for n = 1..1000 (a(1) to a(100) from Harvey P. Dale)

Example

Divisors >1 of 259: 7, 37 and 259, which have all three 1's in binary: 7->'111', 37->'100101' and 259->'100000011', therefore 259 is a term.

Maple

A000120:= proc(n) convert(convert(n, base, 2), `+`) end proc:
filter:= proc(n) local t, f;
if isprime(n) then return false fi;
if n::even then return evalb(n = 2^ilog2(n)) fi;
if n mod 3 = 0 then return evalb(n = 9) fi;
t:= A000120(n);
for f in numtheory:-divisors(n) minus {1, n} do
  if A000120(f) <> t then return false fi;
od;
true
end proc:
select(filter, [$4..10^5]); # Robert Israel, Dec 01 2015

Mathematica

dn1Q[n_]:=!PrimeQ[n]&&Length[Union[(DigitCount[#, 2, 1]&/@Rest[Divisors[ n]])]] == 1; Select[Range[26000], dn1Q] (* Harvey P. Dale, Oct 03 2013 *)

Prog

(PARI) isok(n) = {if (isprime(n) || n==1, return (0), my(nb = norml2(binary(n))); fordiv(n, d, if (d!=1 && norml2(binary(d)) != nb, return (0))); return (1); ); }  \\ Michel Marcus, Jun 05 2013

Crossrefs

Cf. A007088, A002808.

Sequence in context: A272758 A227645 A285438 * A340093 A227243 A272575

Adjacent sequences: A089039 A089040 A089041 * A089043 A089044 A089045

Keyword

nonn,base

Author

Reinhard Zumkeller, Dec 02 2003

Status

approved