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A089042 Composite numbers such that all divisors >1 have the same number of 1's in binary representation. 2
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 (list; graph; refs; listen; history; text; internal format)
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

isok(n) = {if (isprime(n), return (0), 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

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Last modified May 18 03:03 EDT 2021. Contains 343994 sequences. (Running on oeis4.)