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

Author

Reinhard Zumkeller, Dec 02 2003

Keywords

• nonn
• base

Comments

A000120(d)=constant for all d with 1

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)

Examples

			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.
		

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)

Crossrefs

Cf. A007088, A002808.

Programs

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 *)

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

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