A133457 -id:A133457 - cp's OEIS frontend

A342639 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = g(f(n) + f(k)) where g maps the complement, say s, of a finite set of nonnegative integers to the value Sum_{e >= 0 not in s} 2^e, f is the inverse of g, and "+" denotes the Minkowski sum.

0, 1, 1, 0, 3, 0, 3, 1, 1, 3, 0, 7, 2, 7, 0, 1, 1, 3, 3, 1, 1, 0, 3, 0, 15, 0, 3, 0, 7, 1, 5, 3, 3, 5, 1, 7, 0, 15, 2, 7, 0, 7, 2, 15, 0, 1, 1, 7, 3, 1, 1, 3, 7, 1, 1, 0, 3, 0, 31, 4, 11, 4, 31, 0, 3, 0, 3, 1, 1, 3, 7, 5, 5, 7, 3, 1, 1, 3, 0, 7, 2, 7, 0, 15, 6, 15, 0, 7, 2, 7, 0

Offset: 0

Rémy Sigrist, Mar 17 2021

In other words:
- we consider the set S of sets s of nonnegative integers whose complement is finite,
- the function g encodes the "missing integers" in binary:
          g(A001477 \ {1, 4}) = 2^1 + 2^4 = 18
- the function f is the inverse of g:
          f(42) = f(2^1 + 2^3 + 2^5) = A001477 \ {1, 3, 5},
- the Minkowski sum of two sets, say U and V, is the set of sums u+v where u belongs to U and v belongs to V,
- the Minkowski sum is stable over S,
- and T provides an encoding for this operation.
This sequence has connections with A067138; here we consider complements of finite sets of nonnegative integers, there finite sets of nonnegative integers.

			Array T(n, k) begins:
  n\k|   0   1   2   3   4   5   6    7   8   9  10  11  12  13  14   15
  ---+------------------------------------------------------------------
    0|   0   1   0   3   0   1   0    7   0   1   0   3   0   1   0   15
    1|   1   3   1   7   1   3   1   15   1   3   1   7   1   3   1   31
    2|   0   1   2   3   0   5   2    7   0   1   2  11   0   5   2   15
    3|   3   7   3  15   3   7   3   31   3   7   3  15   3   7   3   63
    4|   0   1   0   3   0   1   4    7   0   1   0   3   0   9   4   15
    5|   1   3   5   7   1  11   5   15   1   3   5  23   1  11   5   31
    6|   0   1   2   3   4   5   6    7   0   9   2  11   4  13   6   15
    7|   7  15   7  31   7  15   7   63   7  15   7  31   7  15   7  127
    8|   0   1   0   3   0   1   0    7   0   1   0   3   0   1   8   15
    9|   1   3   1   7   1   3   9   15   1   3   1   7   1  19   9   31
   10|   0   1   2   3   0   5   2    7   0   1  10  11   0   5  10   15
   11|   3   7  11  15   3  23  11   31   3   7  11  47   3  23  11   63
   12|   0   1   0   3   0   1   4    7   0   1   0   3   8   9  12   15
   13|   1   3   5   7   9  11  13   15   1  19   5  23   9  27  13   31
   14|   0   1   2   3   4   5   6    7   8   9  10  11  12  13  14   15
   15|  15  31  15  63  15  31  15  127  15  31  15  63  15  31  15  255

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Wikipedia, Minkowski addition

Cf. A038712, A067138, A133457, A135481, A342640, A342641, A342642.

T(n,k) = { my (v=0); for (x=0, #binary(n)+#binary(k), my (f=0); for (y=0, x, if (!bittest(n,y) && !bittest(k,x-y), f=1; break)); if (!f, v+=2^x)); return (v) }

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, 0) = A135481(n).
T(n, 1) = A038712(n+1).
T(2^n-1, 2^k-1) = 2^(n+k)-1.
T(n, n) = A342640(n).

A342640 a(n) = A342639(n, n).

Original entry on oeis.org

0, 3, 2, 15, 0, 11, 6, 63, 0, 3, 10, 47, 8, 27, 14, 255, 0, 3, 2, 15, 0, 43, 22, 191, 0, 35, 10, 111, 24, 59, 30, 1023, 0, 3, 2, 15, 0, 11, 38, 63, 0, 3, 42, 175, 8, 91, 46, 767, 0, 3, 2, 143, 32, 43, 54, 447, 32, 99, 42, 239, 56, 123, 62, 4095, 0, 3, 2, 15, 0

Offset: 0

Rémy Sigrist, Mar 17 2021

For any n >= 0:
- let s(n) be the unique finite set of nonnegative integers such that n = Sum_{e in s(n)} 2^e,
- then s(a(n)) corresponds to the set of nonnegative integers that are not the sum of two nonnegative integers not in s(n).

			The first terms, alongside the corresponding sets, are:
  n   a(n)  s(n)          s(a(n))
  --  ----  ------------  ------------------------
   0     0  {}            {}
   1     3  {0}           {0, 1}
   2     2  {1}           {1}
   3    15  {0, 1}        {0, 1, 2, 3}
   4     0  {2}           {}
   5    11  {0, 2}        {0, 1, 3}
   6     6  {1, 2}        {1, 2}
   7    63  {0, 1, 2}     {0, 1, 2, 3, 4, 5}
   8     0  {3}           {}
   9     3  {0, 3}        {0, 1}
  10    10  {1, 3}        {1, 3}
  11    47  {0, 1, 3}     {0, 1, 2, 3, 5}
  12     8  {2, 3}        {3}
  13    27  {0, 2, 3}     {0, 1, 3, 4}
  14    14  {1, 2, 3}     {1, 2, 3}
  15   255  {0, 1, 2, 3}  {0, 1, 2, 3, 4, 5, 6, 7}

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A133457, A342639, A342641, A342642.

a(n) = { my (v=0); for (x=0, 2*#binary(n), my (f=0); for (y=0, x, if (!bittest(n,y) && !bittest(n,x-y), f=1; break)); if (!f, v+=2^x)); return (v) }

a(2^n-1) = 4^n-1.

A342641 Numbers k such that A342640(k) = k.

Original entry on oeis.org

0, 2, 6, 10, 14, 22, 30, 38, 42, 46, 54, 62, 78, 94, 110, 126, 142, 150, 158, 170, 174, 182, 190, 206, 222, 238, 254, 286, 310, 318, 350, 382, 414, 438, 446, 478, 510, 542, 558, 574, 606, 622, 638, 670, 682, 686, 702, 734, 750, 766, 798, 830, 862, 894, 926

Offset: 1

Rémy Sigrist, Mar 17 2021

All terms are even.
For any m >= 0:
- let s(m) be the unique finite set of nonnegative integers such that m = Sum_{e in s(m)} 2^e,
- this sequence contains the numbers k such that s(k) is the set of nonnegative integers that are not the sum of two nonnegative integers not in s(k).

			The first terms, alongside the corresponding sets, are:
  n   a(n)  s(a(n))
  --  ----  ---------------
   1     0  {}
   2     2  {1}
   3     6  {1, 2}
   4    10  {1, 3}
   5    14  {1, 2, 3}
   6    22  {1, 2, 4}
   7    30  {1, 2, 3, 4}
   8    38  {1, 2, 5}
   9    42  {1, 3, 5}
  10    46  {1, 2, 3, 5}
  11    54  {1, 2, 4, 5}
  12    62  {1, 2, 3, 4, 5}
  13    78  {1, 2, 3, 6}
  14    94  {1, 2, 3, 4, 6}
  15   110  {1, 2, 3, 5, 6}

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A133457, A342639, A342640, A342642.

is(n) = { my (v=0); for (x=0, 2*#binary(n), my (f=0); for (y=0, x, if (!bittest(n,y) && !bittest(n,x-y), f=1; break)); if (!f, v+=2^x)); return (v==n) }

A342642 Numbers k such that A342640(k) = 0.

Original entry on oeis.org

0, 4, 8, 16, 20, 24, 32, 36, 40, 48, 64, 68, 72, 80, 84, 88, 96, 100, 104, 112, 128, 132, 136, 144, 148, 152, 160, 164, 168, 176, 192, 196, 200, 208, 216, 224, 228, 256, 260, 264, 272, 276, 280, 288, 292, 296, 304, 320, 324, 328, 336, 340, 344, 352, 356, 360

Offset: 1

Rémy Sigrist, Mar 17 2021

For any m >= 0:
- let s(m) be the unique finite set of nonnegative integers such that m = Sum_{e in s(m)} 2^e,
- this sequence contains the numbers k such that every nonnegative integer is the sum of two nonnegative integers not in s(k).
All terms are even.

			The first terms, alongside the corresponding sets, are:
  n   a(n)  s(a(n))
  --  ----  ---------
   1     0  {}
   2     4  {2}
   3     8  {3}
   4    16  {4}
   5    20  {2, 4}
   6    24  {3, 4}
   7    32  {5}
   8    36  {2, 5}
   9    40  {3, 5}
  10    48  {4, 5}
  11    64  {6}
  12    68  {2, 6}
  13    72  {3, 6}
  14    80  {4, 6}
  15    84  {2, 4, 6}

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A133457, A342639, A342640, A342641.

is(n) = { my (v=0); for (x=0, 2*#binary(n), my (f=0); for (y=0, x, if (!bittest(n, y) && !bittest(n, x-y), f=1; break)); if (!f, v+=2^x)); return (v==0) }

A264613 Numbers n such that the Shevelev polynomial {m, n} has a root at m = -1.

Original entry on oeis.org

2, 5, 8, 11, 23, 32, 47, 95, 128, 191, 383, 512, 767, 1535, 2048, 3071, 6143, 8192, 12287, 24575, 32768, 49151, 98303, 131072, 196607, 393215, 524288, 786431, 1572863, 2097152, 3145727, 6291455, 8388608

Offset: 1

N. J. A. Sloane, Nov 28 2015

nonn

From Peter J. C. Moses, Dec 12 2015: (Start)
This appears to split into 3 sequences:
b(n) = 3*4^(n-1)-1, n>=1: 2,11,47,191,767,3071,12287,49151,...,
c(n) = 3*2^(2*n-1)-1, n>=1: 5,23,95,383,1535,6143,24575,98303,...,
d(n) = 2^(2*n+1), n>=1: 8,32,128,512,2048,8192,32768,...;
If this is true, then the next few terms of the sequence are 12582911, 25165823, 33554432, 50331647, 100663295, ...
(End)

Vladimir Shevelev, On the Basis Polynomials in the Theory of Permutations with Prescribed Up-Down Structure, arXiv:0801.0072 [math.CO], 2007-2010. See Appendix.
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. (See Section 11, Problem 3.)
V. Shevelev and J. Spilker, Up-down coefficients for permutations, Elemente der Mathematik, Vol. 68 (2013), no.3, 115-127.

Cf. A133457 (positive integer roots of {m,n}), A263848.

upDown[n_, k_] := upDown[n, k] = Module[{t, m}, t = Flatten[ Reverse[ Position[ Reverse[ IntegerDigits[k, 2]], 1]]]; m = Length[t]; (-1)^m + Sum[upDown[t[[j]], k - 2^(t[[j]] - 1)]*Binomial[n, t[[j]]], {j, 1, m}]];
Reap[For[k = 2, k <= 2^15, k++, If[(upDown[n, k] /. n -> -1) == 0, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Sep 06 2018 *)

Conjectured g.f.: (2 + x*(5 + x*(8 + x*(1 + (-2 - 8*x)* x)))) / (1 + x^3*(-5 + 4*x^3)). - Peter J. C. Moses, Dec 12 2015

More terms (starting at a(6)) from Peter J. C. Moses, Dec 12 2015

A341288 Square array T(n, k), read by antidiagonals, n, k >= 0; T(n, k) = XOR_{u in B(n), v in B(k)} 2^(u XOR v) where XOR denotes the bitwise XOR operator and B(n) gives the exponents in expression for n as a sum of powers of 2.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 1, 3, 0, 0, 4, 3, 3, 4, 0, 0, 5, 8, 0, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 9, 15, 1, 15, 9, 7, 0, 0, 8, 11, 15, 5, 5, 15, 11, 8, 0, 0, 9, 4, 12, 9, 0, 9, 12, 4, 9, 0, 0, 10, 6, 12, 13, 15, 15, 13, 12, 6, 10, 0

Offset: 0

Rémy Sigrist, Feb 08 2021

For any x >= 0, the function n -> T(n, 2^x) is a self-inverse permutation of the nonnegative integers.
The set of nonnegative integers equipped with T forms a commutative monoid; its invertible elements are the odious numbers (A000069).
Hence A000069 equipped with T forms a group.

			Array T(n, k) begins:
  n\k|  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+---------------------------------------------------------------
    0|  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
    1|  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    2|  0   2   1   3   8  10   9  11   4   6   5   7  12  14  13  15 -> A057300
    3|  0   3   3   0  12  15  15  12  12  15  15  12   0   3   3   0
    4|  0   4   8  12   1   5   9  13   2   6  10  14   3   7  11  15 -> A126006
    5|  0   5  10  15   5   0  15  10  10  15   0   5  15  10   5   0
    6|  0   6   9  15   9  15   0   6   6   0  15   9  15   9   6   0
    7|  0   7  11  12  13  10   6   1  14   9   5   2   3   4   8  15
    8|  0   8   4  12   2  10   6  14   1   9   5  13   3  11   7  15
    9|  0   9   6  15   6  15   0   9   9   0  15   6  15   6   9   0
   10|  0  10   5  15  10   0  15   5   5  15   0  10  15   5  10   0
   11|  0  11   7  12  14   5   9   2  13   6  10   1   3   8   4  15
   12|  0  12  12   0   3  15  15   3   3  15  15   3   0  12  12   0
   13|  0  13  14   3   7  10   9   4  11   6   5   8  12   1   2  15
   14|  0  14  13   3  11   5   6   8   7   9  10   4  12   2   1  15
   15|  0  15  15   0  15   0   0  15  15   0   0  15   0  15  15   0
                                                                     \
                                                                      v
                                                                    A010060

Rémy Sigrist, Table of n, a(n) for n = 0..10010 (rows 0..140)
Rémy Sigrist, Colored representation of the table for n, k < 2^10
Rémy Sigrist, Colored representation of the table over the first 128 odious numbers
Rémy Sigrist, Colored representation of the table over the first 1024 evil numbers (white pixels correspond to 0's)

Cf. A000069, A010060, A057300, A126006, A133457.

B(n) = { my (b=vector(hammingweight(n))); for (k=1, #b, n -= 2^(b[k] = valuation(n, 2))); b }
T(n,k) = { my (nn=B(n), kk=B(k), v=0); for (i=1, #nn, for (j=1, #kk, v=bitxor(v, 2^bitxor(nn[i], kk[j])))); v }

T(n, k) = T(k, n) (T is commutative).
T(m, T(n, k)) = T(T(m, n), k) (T is associative).
T(n, 0) = 0 (0 is an absorbing element for T).
T(n, 1) = n (1 is the neutral element for T).
T(n, 2) = A057300(n).
T(n, 4) = A126006(n).
T(n, n) = A010060(n).
A010060(T(n, k)) = A010060(n) * A010060(k).

A263848 Irregular triangle read by rows: row n gives coefficients of basis polynomial {n,k} expressed in terms of binomial coefficients, high order terms first.

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 1, -1, 1, 1, 0, 0, -1, 2, 0, -1, 1, 2, -1, 0, 1, 1, -1, 1, -1, 1, 0, 0, 0, -1, 3, 0, 0, -1, 1, 5, 0, -1, 0, 1, 3, 0, -1, 1, -1, 3, -1, 0, 0, 1, 5, -2, 0, 1, -1, 3, -2, 1, 0, -1, 1, -1, 1, -1, 1, 1, 0, 0, 0, 0, -1, 4, 0, 0, 0, -1, 1, 9, 0, 0

Offset: 0

N. J. A. Sloane, Nov 15 2015

			Triangle begins:
  1,
  1, -1,
  1,  0, -1,
  1, -1,  1,
  1,  0,  0, -1,
  2,  0, -1,  1,
  2, -1,  0,  1,
  1, -1,  1, -1,
  1,  0,  0,  0, -1,
  3,  0,  0, -1,  1,
  ...

Peter J. C. Moses, First 300 rows.
Vladimir Shevelev, On the Basis Polynomials in the Theory of Permutations with Prescribed Up-Down Structure, arXiv|math.CO/0801.0072, 2007-2010. See Appendix.
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1.
V. Shevelev and J. Spilker, Up-down coefficients for permutations, Elemente der Mathematik, Vol. 68 (2013), no. 3, 115-127.

Cf. A133457, A264613.

More terms from Peter J. C. Moses, Dec 12 2015

A293390 Least m such that the exponents in expression for n as a sum of distinct powers of 2 are pairwise distinct mod m; a(0) = 0 by convention.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 3, 1, 2, 3, 4, 2, 4, 3, 4, 1, 3, 2, 5, 3, 3, 4, 5, 2, 5, 4, 5, 3, 5, 4, 5, 1, 2, 3, 3, 2, 4, 5, 6, 3, 4, 3, 6, 4, 4, 5, 6, 2, 3, 5, 6, 4, 6, 5, 6, 3, 6, 5, 6, 4, 6, 5, 6, 1, 4, 2, 4, 3, 5, 3, 7, 2, 4, 4, 4, 5, 5, 6, 7, 3, 5, 4, 7, 3, 5, 6

Offset: 0

Rémy Sigrist, Oct 08 2017

The set of exponents in expression for n as a sum of distinct powers of 2 corresponds to the n-th row of A133457.
The sum of digits of n in base 2^a(n), say s, can be computed without carry in base 2; the Hamming weight of s equals the Hamming weight of n.
a(n) >= A000120(n) for any n > 0.
Apparently, a(n) = A000120(n) iff n = 0 or n belongs to A100290.
a(n) <= A070939(n) for any n >= 0.
For any sequence s of distinct nonnegative integers (s(n) being defined for n >= 0):
- let D_s be defined for any n > 0 by D_s(n) = a(Sum_{k=0..n-1} 2^s(k)),
- then D_s is the discriminator of s as introduced by Arnold, Benkoski, and McCabe in 1985,
- D_s(1) = 1,
- D_s(n) >= n for any n >= 1,
- D_s(n+1) >= D_s(n) for any n >= 1.

			For n=42:
- 42 = 2^5 + 2^3 + 2^1,
- 5 mod 1 = 3 mod 1,
- 5 mod 2 = 3 mod 2,
- 5 mod 3, 3 mod 3 and 1 mod 3 are all distinct,
- hence a(42) = 3.

Robert Israel, Table of n, a(n) for n = 0..10000
Sajed Haque, Jeffrey Shallit, Discriminators and k-Regular Sequences, arXiv:1605.00092 [cs.DM], 2016.

Cf. A000041, A000045, A000058, A000069, A000108, A000110, A000120, A000215, A001147, A001566, A001969, A002808, A003095, A005823, A016726, A062383, A070939, A076793, A100290, A133457, A173195, A270097, A270151, A270176, A272633, A272881, A272882, A273037, A273041, A273043, A273044, A273056, A273062, A273064, A273068, A273237, A273377

f:= proc(n) local L,D,k;
  L:= convert(n,base,2);
  L:= select(t -> L[t+1]=1, [$0..nops(L)-1]);
  if nops(L) = 1 then return 1 fi;
  D:= {seq(seq(L[j]-L[i],i=1..j-1),j=2..nops(L))};
  D:= `union`(seq(numtheory:-divisors(i),i=D));
  min({$2..max(D)+1} minus D)
end proc:
0, seq(f(i),i=1..100); # Robert Israel, Oct 08 2017

{0}~Join~Table[Function[r, SelectFirst[Range@ 10, Length@ Union@ Mod[r, #] == Length@ r &]][Join @@ Position[#, 1] - 1 &@ Reverse@ IntegerDigits[n, 2]], {n, 86}] (* Michael De Vlieger, Oct 08 2017 *)

a(n) = if (n, my (d=Vecrev(binary(n)), x = []); for (i=1, #d, if (d[i], x = concat(x, i-1))); for (m=1, oo, if (#Set(vector(#x, i, x[i]%m))==#x, return (m))), return (0))

a(2*n) = a(n) for any n >= 0.
a(2^k-1) = k for any k >= 0.
a(n) = 1 iff n = 2^k for some k >= 0.
a(n) = 2 iff n belongs to A173195.
a(Sum_{k=1..n} 2^(k^2)) = A016726(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000069(k)) = A062383(n) for any n >= 1.
a(Sum_{k=0..n} 2^(2^k)) = A270097(n) for any n >= 0.
a(Sum_{k=1..n} 2^A000045(k+1)) = A270151(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000041(k)) = A270176(n) for any n >= 1.
a(A076793(n)) = A272633(n) for any n >= 0.
a(Sum_{k=1..n} 2^A001969(k)) = A272881(n) for any n >= 1.
a(Sum_{k=1..n} 2^A005823(k)) = A272882(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000215(k-1)) = A273037(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000108(k)) = A273041(n) for any n >= 1.
a(Sum_{k=1..n} 2^A001566(k)) = A273043(n) for any n >= 1.
a(Sum_{k=1..n} 2^A003095(k)) = A273044(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000058(k-1)) = A273056(n) for any n >= 1.
a(Sum_{k=1..n} 2^A002808(k)) = A273062(n) for any n >= 1.
a(Sum_{k=1..n} 2^(k!)) = A273064(n) for any n >= 1.
a(Sum_{k=1..n} 2^(k^k)) = A273068(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000110(k)) = A273237(n) for any n >= 1.
a(Sum_{k=1..n} 2^A001147(k)) = A273377(n) for any n >= 1.

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.

Original entry on oeis.org

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

Rémy Sigrist, Oct 12 2017

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.

			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.

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

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

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

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)); }

A293664 For n >= 0, let E_n be the set of exponents in expression of 2n as a sum of distinct powers of 2 (2n = Sum_{e in E_n} 2^e); a(n) = number of distinct values taken by the expression Sum_{e in E_n} s(e)*2^e when s runs over all functions from the positive numbers to the set { +1, -1 }.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 4, 7, 2, 4, 4, 8, 4, 7, 8, 11, 2, 4, 4, 8, 4, 8, 7, 12, 4, 7, 8, 13, 8, 12, 13, 16, 2, 4, 4, 8, 4, 8, 8, 13, 4, 8, 7, 14, 8, 13, 14, 17, 4, 7, 8, 13, 8, 14, 13, 18, 8, 12, 14, 19, 15, 18, 19, 22, 2, 4, 4, 8, 4, 8, 8, 14, 4, 8, 8, 15, 7, 12

Offset: 0

Rémy Sigrist, Oct 14 2017

More informally, any number n encodes a finite sets of positive numbers, say { e_1, e_2, ..., e_h }, and a(n) gives the number of distinct values of the form +- e_1 +- e_2 ... +- e_h.
The set of exponents in expression for n as a sum of distinct powers of 2 corresponds to the n-th row of A133457.
A number n belongs to A293576 iff a(n) is odd.
a(n) <= 2^A000120(n) for any n >= 0.

			For n = 15:
- E_15 = { 1, 2, 3, 4 },
- the possible "plus-minus" sums are:
  +4 +3 +2 +1 =  10   (1st value)
  +4 +3 +2 -1 =   8   (2nd value)
  +4 +3 -2 +1 =   6   (3rd value)
  +4 +3 -2 -1 =   4   (4th value)
  +4 -3 +2 +1 =   4   (already seen)
  +4 -3 +2 -1 =   2   (5th value)
  +4 -3 -2 +1 =   0   (6th value)
  +4 -3 -2 -1 =  -2   (7th value)
  -4 +3 +2 +1 =   2   (already seen)
  -4 +3 +2 -1 =   0   (already seen)
  -4 +3 -2 +1 =  -2   (already seen)
  -4 +3 -2 -1 =  -4   (8th value)
  -4 -3 +2 +1 =  -4   (already seen)
  -4 -3 +2 -1 =  -6   (9th value)
  -4 -3 -2 +1 =  -8   (10th value)
  -4 -3 -2 -1 = -10   (11th value)
- hence, a(15) = 11.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A133457, A293576.

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

cp's OEIS Frontend

A342639 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = g(f(n) + f(k)) where g maps the complement, say s, of a finite set of nonnegative integers to the value Sum_{e >= 0 not in s} 2^e, f is the inverse of g, and "+" denotes the Minkowski sum.

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A342640 a(n) = A342639(n, n).

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A342641 Numbers k such that A342640(k) = k.

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A342642 Numbers k such that A342640(k) = 0.

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A264613 Numbers n such that the Shevelev polynomial {m, n} has a root at m = -1.

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A341288 Square array T(n, k), read by antidiagonals, n, k >= 0; T(n, k) = XOR_{u in B(n), v in B(k)} 2^(u XOR v) where XOR denotes the bitwise XOR operator and B(n) gives the exponents in expression for n as a sum of powers of 2.

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A263848 Irregular triangle read by rows: row n gives coefficients of basis polynomial {n,k} expressed in terms of binomial coefficients, high order terms first.

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A293390 Least m such that the exponents in expression for n as a sum of distinct powers of 2 are pairwise distinct mod m; a(0) = 0 by convention.

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A293664 For n >= 0, let E_n be the set of exponents in expression of 2n as a sum of distinct powers of 2 (2n = Sum_{e in E_n} 2^e); a(n) = number of distinct values taken by the expression Sum_{e in E_n} s(e)*2^e when s runs over all functions from the positive numbers to the set { +1, -1 }.