A342641 -id:A342641 - 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.

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) }

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

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