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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

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

EXAMPLE

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.

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

PROG

(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)); }

CROSSREFS

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

Sequence in context: A053696 A090503 A059520 * A233301 A274255 A233747

Adjacent sequences:  A293573 A293574 A293575 * A293577 A293578 A293579

KEYWORD

nonn,base

AUTHOR

Rémy Sigrist, Oct 12 2017

STATUS

approved

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Last modified September 20 20:12 EDT 2019. Contains 327247 sequences. (Running on oeis4.)