A327188 -id:A327188 - cp's OEIS frontend

A327186 For any n >= 0: consider the different ways to split the binary representation of n into two (possibly empty) parts, say with value x and y; a(n) is the least possible value of x OR y (where OR denotes the bitwise OR operator).

0, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 3, 3, 3, 3, 3, 1, 1, 2, 3, 5, 5, 6, 7, 3, 3, 3, 3, 7, 7, 7, 7, 1, 1, 2, 3, 4, 5, 6, 7, 5, 5, 7, 7, 5, 5, 7, 7, 3, 3, 3, 3, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 1, 1, 2, 3, 4, 5, 6, 7, 9, 9, 10, 11, 12, 13, 14, 15, 5, 5, 7, 7, 5

Offset: 0

Rémy Sigrist, Aug 25 2019

			For n=42:
- the binary representation of 42 is "101010",
- there are 7 ways to split it:
   - "" and "101010": x=0 and y=42: 0 OR 42 = 42,
   - "1" and "01010": x=1 and y=10: 1 OR 10 = 11,
   - "10" and "1010": x=2 and y=10: 2 OR 10 = 10,
   - "101" and "010": x=5 and y=2: 5 OR 2 = 7,
   - "1010" and "10": x=10 and y=2: 10 OR 2 = 10,
   - "10101" and "0": x=21 and y=0: 21 OR 0 = 21,
   - "101010" and "": x=42 and y=0: 42 OR 0 = 42,
- hence a(42) = 7.

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

Cf. A327187 (x XOR y variant), A327188 (x AND y variant).
Cf. A327189 (x + y variant), A327190 (x * y variant), A327191 (x - y variant).
Cf. A327192 (max(x, y) variant), A327193 (min(x, y) variant).
Cf. A327194 (x^2 + y^2 variant), A327195 (x^2 - y^2 variant).

a(n) = my (v=oo, b=binary(n)); for (w=0, #b, v=min(v, bitor(fromdigits(b[1..w],2), fromdigits(b[w+1..#b],2)))); v

a(n) = 1 iff n = 2^k or n = 2^k + 1 for some k >= 0.

A330955 For any n >= 0: consider all pairs of numbers (x, y) whose binary representations can be interleaved (or shuffled) to produce the binary representation of n (possibly with leading zeros); a(n) is the greatest possible value of x AND y (where AND denotes the bitwise AND operator).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Jan 04 2020

			For n = 5:
- the binary representation of 5 is "101",
- the possible values for (x, y), restricted to x >= y without loss of generality, are:
  bin(5)   x  y  x AND y
  -------  -  -  -------
  "101"    5  0        0
  "1/01"   1  1        1
  "10/1"   2  1        0
  "1/0/1"  3  0        0
- hence a(5) = 1.

Rémy Sigrist, C program for A330955
Index entries for sequences related to binary expansion of n

See A330925 for similar sequences.

Cf. A327188.

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cp's OEIS Frontend

A327186 For any n >= 0: consider the different ways to split the binary representation of n into two (possibly empty) parts, say with value x and y; a(n) is the least possible value of x OR y (where OR denotes the bitwise OR operator).

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