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A293576
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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.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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PROG
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(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)); }
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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