A327186 -id:A327186 - cp's OEIS frontend

A330925 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*y.

0, 0, 0, 1, 0, 2, 2, 3, 0, 4, 4, 6, 4, 6, 6, 9, 0, 8, 8, 12, 8, 12, 12, 15, 8, 12, 12, 18, 12, 18, 18, 21, 0, 16, 16, 24, 16, 24, 24, 28, 16, 24, 24, 30, 24, 30, 30, 35, 16, 24, 24, 36, 24, 36, 36, 42, 24, 36, 36, 42, 36, 42, 42, 49, 0, 32, 32, 48, 32, 48, 48

Offset: 0

Rémy Sigrist, Jan 03 2020

Interleaving (or shuffling) two strings means combining all their characters while preserving the order of all characters in individual strings; for example, "12345" is the interleaving of "14" and "235".

			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*y
  -------  -  -  ---
  "101"    5  0    0
  "1/01"   1  1    1
  "10/1"   2  1    2
  "1/0/1"  3  0    0
- hence a(5) = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, C program for A330925
Index entries for sequences related to binary expansion of n

Cf. A330955 (x AND y variant), A330958 (x OR y variant), A330959 (x XOR y variant).
Cf. A330956 (min(x, y) variant), A330957 (max(x, y) variant).
Cf. A330960 (x + y variant), A330961 (x - y variant).
Cf. A330962 (x^2 + y^2 variant), A330963 (x^2 - y^2 variant).
See A327186 for similar sequences where we split the binary representation.

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a(n) = 0 iff n is zero or a power of 2.

a(2*n) = 2*a(n).

A327187 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 XOR y (where XOR denotes the bitwise XOR operator).

Original entry on oeis.org

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

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 XOR 42 = 42,
   - "1" and "01010": x=1 and y=10: 1 XOR 10 = 11,
   - "10" and "1010": x=2 and y=10: 2 XOR 10 = 8,
   - "101" and "010": x=5 and y=2: 5 XOR 2 = 7,
   - "1010" and "10": x=10 and y=2: 10 XOR 2 = 8,
   - "10101" and "0": x=21 and y=0: 21 XOR 0 = 21,
   - "101010" and "": x=42 and y=0: 42 XOR 0 = 42,
- hence a(42) = 7.

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

See A327186 for other variants.

Cf. A175468.

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

a(n) = 0 iff n = 0 or n belongs to A175468.

A327188 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 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, 0, 1, 2, 3, 0, 1, 2, 2, 0, 1, 2, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 2, 4, 4, 4, 4, 0, 1, 2, 2, 4, 5, 4, 5, 0, 1, 2, 3, 4, 4, 6, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 2, 4, 4, 4, 4, 0, 1, 2, 2, 4, 4, 4, 4, 0, 1, 2, 2, 4, 5, 4

Offset: 0

Rémy Sigrist, Aug 25 2019

The first 10000 positive integers where the sequence equals zero match the first 10000 terms of A082662; is that true forever?

			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 AND 42 = 0,
   - "1" and "01010": x=1 and y=10: 1 AND 10 = 0,
   - "10" and "1010": x=2 and y=10: 2 AND 10 = 2,
   - "101" and "010": x=5 and y=2: 5 AND 2 = 0,
   - "1010" and "10": x=10 and y=2: 10 AND 2 = 2,
   - "10101" and "0": x=21 and y=0: 21 AND 0 = 0,
   - "101010" and "": x=42 and y=0: 42 AND 0 = 0,
- hence a(42) = 2.

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

See A327186 for other variants.

Cf. A082662.

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

A327189 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 + y.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 10, 11, 12, 3, 4, 5, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 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 + 42 = 42,
   - "1" and "01010": x=1 and y=10: 1 + 10 = 11,
   - "10" and "1010": x=2 and y=10: 2 + 10 = 12,
   - "101" and "010": x=5 and y=2: 5 + 2 = 7,
   - "1010" and "10": x=10 and y=2: 10 + 2 = 12,
   - "10101" and "0": x=21 and y=0: 21 + 0 = 21,
   - "101010" and "": x=42 and y=0: 42 + 0 = 42,
- hence a(42) = 7.

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

See A327186 for other variants.

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

a(n) = 1 iff n is a power of 2.

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

A327191 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 abs(x - y).

Original entry on oeis.org

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

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: abs(0 - 42) = 42,
   - "1" and "01010": x=1 and y=10: abs(1 - 10) = 9,
   - "10" and "1010": x=2 and y=10: abs(2 - 10) = 8,
   - "101" and "010": x=5 and y=2: abs(5 - 2) = 3,
   - "1010" and "10": x=10 and y=2: abs(10 - 2) = 8,
   - "10101" and "0": x=21 and y=0: abs(21 - 0) = 21,
   - "101010" and "": x=42 and y=0: abs(42 - 0) = 42,
- hence a(42) = 3.

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

See A327186 for other variants.

Cf. A175468.

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

a(n) = 0 iff n = 0 or n belongs to A175468.

A327192 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 max(x, y).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 3, 3, 3, 3, 3, 1, 1, 2, 3, 4, 5, 5, 5, 3, 3, 3, 3, 4, 5, 6, 7, 1, 1, 2, 3, 4, 5, 6, 7, 5, 5, 5, 5, 5, 5, 6, 7, 3, 3, 3, 3, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9, 5, 5, 5, 5, 5, 5, 6

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: max(0, 42) = 42,
   - "1" and "01010": x=1 and y=10: max(1, 10) = 10,
   - "10" and "1010": x=2 and y=10: max(2, 10) = 10,
   - "101" and "010": x=5 and y=2: max(5, 2) = 5,
   - "1010" and "10": x=10 and y=2: max(10, 2) = 10,
   - "10101" and "0": x=21 and y=0: max(21, 0) = 21,
   - "101010" and "": x=42 and y=0: max(42, 0) = 42,
- hence a(42) = 5.

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

See A327186 for other variants.

a(n) = my (v=oo, b=binary(n)); for (w=0, #b, v=min(v, max(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.

A327193 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 greatest possible value of min(x, y).

Original entry on oeis.org

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

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: min(0, 42) = 0,
   - "1" and "01010": x=1 and y=10: min(1, 10) = 1,
   - "10" and "1010": x=2 and y=10: min(2, 10) = 2,
   - "101" and "010": x=5 and y=2: min(5, 2) = 2,
   - "1010" and "10": x=10 and y=2: min(10, 2) = 2,
   - "10101" and "0": x=21 and y=0: min(21, 0) = 0,
   - "101010" and "": x=42 and y=0: min(42, 0) = 0,
- hence a(42) = 2.

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

See A327186 for other variants.

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

a(n) = 0 iff n = 0 or n is a power of 2.

A327194 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^2 + y^2.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 5, 10, 1, 2, 5, 10, 9, 10, 13, 18, 1, 2, 5, 10, 17, 26, 29, 34, 9, 10, 13, 18, 25, 34, 45, 58, 1, 2, 5, 10, 17, 26, 37, 50, 25, 26, 29, 34, 41, 50, 61, 74, 9, 10, 13, 18, 25, 34, 45, 58, 49, 50, 53, 58, 65, 74, 85, 98, 1, 2, 5, 10, 17, 26, 37

Offset: 0

Rémy Sigrist, Aug 25 2019

This sequence shares graphical features with A286327.

			For n=42:
- the binary representation of 42 is "101010",
- there are 7 ways to split it:
   - "" and "101010": x=0 and y=42: 0^2 + 42^2 = 1764,
   - "1" and "01010": x=1 and y=10: 1^2 + 10^2 = 101,
   - "10" and "1010": x=2 and y=10: 2^2 + 10^2 = 104,
   - "101" and "010": x=5 and y=2: 5^2 + 2^2 = 29,
   - "1010" and "10": x=10 and y=2: 10^2 + 2^2 = 104,
   - "10101" and "0": x=21 and y=0: 21^2 + 0^2 = 441,
   - "101010" and "": x=42 and y=0: 42^2 + 0^2 = 1764,
- hence a(42) = 29.

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

See A327186 for other variants.

Cf. A286327.

Table[Min[Total[#^2]&/@Table[FromDigits[#,2]&/@TakeDrop[IntegerDigits[n,2],d],{d,0,IntegerLength[n,2]}]],{n,0,80}] (* Harvey P. Dale, Mar 03 2023 *)

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

a(n) = 1 iff n is a power of 2.

A327195 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 abs(x^2 - y^2).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 3, 8, 1, 0, 0, 5, 9, 8, 5, 0, 1, 0, 0, 5, 12, 21, 21, 16, 9, 8, 5, 0, 7, 16, 27, 40, 1, 0, 0, 5, 0, 9, 20, 33, 25, 24, 21, 16, 9, 0, 11, 24, 9, 8, 5, 0, 7, 11, 0, 13, 49, 48, 45, 40, 33, 24, 13, 0, 1, 0, 0, 5, 0, 9, 20, 15, 48, 65, 77, 72, 65

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: abs(0^2 - 42^2) = 1764,
   - "1" and "01010": x=1 and y=10: abs(1^2 - 10^2) = 99,
   - "10" and "1010": x=2 and y=10: abs(2^2 - 10^2) = 96,
   - "101" and "010": x=5 and y=2: abs(5^2 - 2^2) = 21,
   - "1010" and "10": x=10 and y=2: abs(10^2 - 2^2) = 96,
   - "10101" and "0": x=21 and y=0: abs(21^2 - 0^2) = 441,
   - "101010" and "": x=42 and y=0: abs(42^2 - 0^2) = 1764,
- hence a(42) = 21.

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

See A327186 for other variants.

Cf. A175468.

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

a(n) = 0 iff n = 0 or n belongs to A175468.

a(n) = 1 iff n is a power of 2.

A327190 For any n > 0: consider the different ways to split the binary representation of 2n+1 into two nonempty parts, say with value x and y; a(n) is the least possible value of x y.

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 7, 1, 3, 5, 7, 3, 9, 7, 15, 1, 3, 5, 7, 5, 11, 11, 15, 3, 9, 13, 21, 7, 21, 15, 31, 1, 3, 5, 7, 9, 11, 13, 15, 5, 15, 21, 23, 11, 27, 23, 31, 3, 9, 15, 21, 13, 33, 27, 45, 7, 21, 29, 49, 15, 45, 31, 63, 1, 3, 5, 7, 9, 11, 13, 15, 9, 19, 21

Offset: 1

Rémy Sigrist, Aug 25 2019

All terms are odd.

			For n=42:
- the binary representation of 85 is "1010101",
- there are 6 ways to split it:
   - "1" and "010101": x=1 and y=21: 1 * 21 = 21,
   - "10" and "10101": x=2 and y=21: 2 * 21 = 42,
   - "101" and "0101": x=5 and y=5: 5 * 5 = 25,
   - "1010" and "101": x=10 and y=5: 10 * 5 = 50,
   - "10101" and "01": x=21 and y=1: 21 * 1 = 21,
   - "101010" and "1": x=42 and y=1: 42 * 1 = 42,
- hence a(42) = 21.

Rémy Sigrist, Table of n, a(n) for n = 1..8192

See A327186 for other variants.

Cf. A000225.

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

a(n) = 1 iff n is a power of 2.

a(n) = n iff n is a positive Mersenne number (A000225). - Bernard Schott, Aug 26 2019

cp's OEIS Frontend

A330925 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*y.

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A327187 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 XOR y (where XOR denotes the bitwise XOR operator).

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A327188 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 greatest possible value of x AND y (where AND denotes the bitwise AND operator).

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A327189 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 + y.

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A327191 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 abs(x - y).

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A327192 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 max(x, y).

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A327193 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 greatest possible value of min(x, y).

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A327194 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^2 + y^2.

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A327190 For any n > 0: consider the different ways to split the binary representation of 2n+1 into two nonempty parts, say with value x and y; a(n) is the least possible value of x y.