A342698 -id:A342698 - cp's OEIS frontend

A342699 Numbers k such that A342698(k) = k.

0, 1, 3, 7, 9, 12, 15, 17, 19, 24, 25, 28, 31, 33, 35, 39, 48, 49, 51, 56, 57, 60, 63, 65, 67, 71, 79, 96, 97, 99, 103, 112, 113, 115, 120, 121, 124, 127, 129, 131, 135, 143, 153, 159, 192, 193, 195, 199, 204, 207, 224, 225, 227, 231, 240, 241, 243, 248, 249

Offset: 1

Rémy Sigrist, Mar 18 2021

Equivalently, these are the numbers k such that each bit in the binary representation of k is next to a bit with the same value (and we consider that the first bit is next to the last bit). Hence, all terms of A033015 belong to this sequence.

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

Cf. A033015, A342698.

is(n) = my (w=#binary(n)); sum(k=0, w-1, ((bittest(n, (k-1)%w)+bittest(n, k%w)+bittest(n, (k+1)%w))>=2) * 2^k)==n

A342697 For any number n with binary expansion Sum_{k >= 0} b(k) * 2^k, the binary expansion of a(n) is Sum_{k >= 0} floor((b(k) + b(k+1) + b(k+2))/2) * 2^k.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 3, 3, 0, 0, 2, 3, 6, 7, 7, 7, 0, 0, 0, 1, 4, 5, 7, 7, 12, 12, 14, 15, 14, 15, 15, 15, 0, 0, 0, 1, 0, 1, 3, 3, 8, 8, 10, 11, 14, 15, 15, 15, 24, 24, 24, 25, 28, 29, 31, 31, 28, 28, 30, 31, 30, 31, 31, 31, 0, 0, 0, 1, 0, 1, 3, 3, 0, 0, 2, 3, 6

Offset: 0

Rémy Sigrist, Mar 18 2021

The value of the k-th bit in a(n) corresponds to the most frequent value in the bit triple starting at the k-th bit in n.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     0      10          0
   3     1      11          1
   4     0     100          0
   5     1     101          1
   6     3     110         11
   7     3     111         11
   8     0    1000          0
   9     0    1001          0
  10     2    1010         10
  11     3    1011         11
  12     6    1100        110
  13     7    1101        111
  14     7    1110        111
  15     7    1111        111

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^20.
Index entries for sequences related to binary expansion of n

Cf. A048715, A048730, A048733, A342698, A342700.

A342697[n_] := Quotient[7*n - BitXor[n, 2*n, 4*n], 8];
Array[A342697, 100, 0] (* Paolo Xausa, Aug 06 2025 *)

a(n) = sum(k=0, #binary(n), ((bittest(n, k)+bittest(n, k+1)+bittest(n, k+2))>=2) * 2^k)

a(n) = 0 iff n belongs to A048715.

a(n) = floor(A048730(n)/8) = floor(A048733(n)/2). - Kevin Ryde, Mar 26 2021

A342700 For any number n with binary expansion (b(1), b(2), ..., b(k)), the binary expansion of a(n) is (1-floor((b(k)+b(1)+b(2))/2), 1-floor((b(1)+b(2)+b(3))/2), ..., 1-floor((b(k-1)+b(k)+b(1))/2)).

Original entry on oeis.org

0, 0, 2, 0, 7, 0, 0, 0, 15, 6, 10, 0, 3, 0, 0, 0, 31, 14, 30, 12, 23, 4, 16, 0, 7, 6, 2, 0, 3, 0, 0, 0, 63, 30, 62, 28, 63, 28, 56, 24, 47, 14, 42, 8, 35, 0, 32, 0, 15, 14, 14, 12, 7, 4, 0, 0, 7, 6, 2, 0, 3, 0, 0, 0, 127, 62, 126, 60, 127, 60, 120, 56, 127, 62

Offset: 0

Rémy Sigrist, Mar 18 2021

This sequence is a variant of A342698; here the value of the k-th bit of a(n) is the less frequent value in the bit triple centered around the k-th bit of n.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     2      10         10
   3     0      11          0
   4     7     100        111
   5     0     101          0
   6     0     110          0
   7     0     111          0
   8    15    1000       1111
   9     6    1001        110
  10    10    1010       1010
  11     0    1011          0
  12     3    1100         11
  13     0    1101          0
  14     0    1110          0
  15     0    1111          0

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A003817, A020988 (fixed points), A342698.

a(n) = my (w=#binary(n)); sum(k=0, w-1, ((bittest(n, (k-1)%w)+bittest(n, k%w)+bittest(n, (k+1)%w))<=1) * 2^k)

a(n) + A342698(n) = A003817(n).

a(n) = n iff n belongs to A020988.

cp's OEIS Frontend

A342699 Numbers k such that A342698(k) = k.

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A342697 For any number n with binary expansion Sum_{k >= 0} b(k) * 2^k, the binary expansion of a(n) is Sum_{k >= 0} floor((b(k) + b(k+1) + b(k+2))/2) * 2^k.

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A342700 For any number n with binary expansion (b(1), b(2), ..., b(k)), the binary expansion of a(n) is (1-floor((b(k)+b(1)+b(2))/2), 1-floor((b(1)+b(2)+b(3))/2), ..., 1-floor((b(k-1)+b(k)+b(1))/2)).

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