A329936 Binary hoax numbers: composite numbers k such that sum of bits of k equals the sum of bits of the distinct prime divisors of k.

4, 8, 9, 15, 16, 32, 45, 49, 50, 51, 55, 64, 75, 85, 100, 117, 126, 128, 135, 153, 159, 162, 171, 185, 190, 200, 205, 207, 215, 222, 225, 238, 246, 249, 252, 253, 256, 287, 303, 319, 324, 333, 338, 350, 369, 374, 378, 380, 400, 407, 438, 442, 444, 469, 471

Offset: 1

Author

Amiram Eldar, Nov 24 2019

Keywords

• nonn
• base

Comments

Analogous to A278909 (binary Smith numbers) as A019506 (hoax numbers) is analogous to A006753 (Smith numbers).

Includes all the powers of 2 except for 1 and 2.

Examples

			4 = 2^2 is in the sequence since the binary representation of 4 is 100 and 1 + 0 + 0 = 1, and the binary representation of 2 is 10 and 1 + 0 = 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006753, A019506, A278909.

Programs

Maple

filter:= proc(n)
if isprime(n) then return false fi;
convert(convert(n,base,2),`+`) = add(convert(convert(t,base,2),`+`),t=numtheory:-factorset(n))
end proc:
select(filter, [$2..1000]); # Robert Israel, Nov 28 2019

Mathematica

binWt[n_] := Total @ IntegerDigits[n, 2]; binHoaxQ[n_] := CompositeQ[n] && Total[binWt /@ FactorInteger[n][[;; , 1]]] == binWt[n]; Select[Range[500], binHoaxQ]

PARI

is(n)= my(f=factor(n)[,1]); sum(i=1,#f, hammingweight(f[i]))==hammingweight(n) && !isprime(n) \\ Charles R Greathouse IV, Nov 28 2019