A342639 -id:A342639 - cp's OEIS frontend

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

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) }

cp's OEIS Frontend

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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