A293576 Numbers n such that the set of exponents in expression for 2*n as a sum of distinct powers of 2 can be partitioned into two parts with equal sums.

0, 7, 13, 15, 22, 25, 27, 30, 39, 42, 45, 47, 49, 51, 54, 59, 60, 62, 75, 76, 82, 85, 87, 90, 93, 95, 97, 99, 102, 107, 108, 110, 117, 119, 120, 122, 125, 127, 141, 143, 147, 148, 153, 155, 158, 162, 165, 167, 170, 173, 175, 179, 180, 185, 187, 188, 190, 193

Offset: 1

Author

Rémy Sigrist, Oct 12 2017

Keywords

• nonn
• base

Comments

More informally, this sequence encodes finite sets of positive numbers, say { e_1, e_2, ..., e_h }, such that +- e_1 +- e_2 ... +- e_h = 0 has a solution.

The set of exponents in expression for n as a sum of distinct powers of 2 corresponds to the n-th row of A133457.

No term can have a Hamming weight of 1 or 2.

If x and y belong to this sequence and x AND y = 0 (where AND stands for the bitwise and-operator), then x + y belongs to this sequence.

If k has an odd Hamming weight, then there are only a finite number of terms with the same odd part as k (see A000265 for the odd part of a number).

The number 2^k-1 belongs to this sequence iff A063865(k) > 0.

If k has Hamming weight > 1, then k + 2^(A029931(k)-1) belongs to this sequence.

Examples

			2*42 = 2^6 + 2^4 + 2^2 and 6 = 4 + 2, hence 42 appears in the sequence.
2*11 = 2^4 + 2^2 + 2^1 and { 1, 2, 4 } cannot be partitioned into two parts with equals sums, hence 11 does not appear in the sequence.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000120, A000265, A029931, A133457, A293664.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, is(t=0),
      (i-> b(n-2^i, t-i) or b(n-2^i, t+i))(ilog2(n)))
    end:
a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, -1, a(n-1)) while not b(2*k, 0) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 22 2017

PARI

is(n) = { my (v=Set(0)); my (b = Vecrev(binary(n))); for (i=1, #b, if (b[i], v = set union(Set(vector(#v, k, v[k]-i)), Set(vector(#v, k, v[k]+i))););); return (set search(v,0)); }