A353158 -id:A353158 - cp's OEIS frontend

A353157 a(n) is the distance from n to the nearest integer whose binary expansion has no common 1-bits with that of n.

0, 1, 1, 1, 1, 3, 2, 1, 1, 3, 5, 5, 4, 3, 2, 1, 1, 3, 5, 7, 9, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25

Offset: 0

Rémy Sigrist, Apr 27 2022

Equivalently the distance to the nearest integer that can be added without carries in base 2.

			For n = 42 ("101010" in binary):
- 21 ("10101") is the greatest number <= 42 that has no common 1-bits with 42,
- 128 ("1000000") is the least number >= 42 that has no common 1-bits with 42,
- so a(42) = min(42-21, 128-42) = min(21, 86) = 21.

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

Cf. A006257, A020988, A097110, A080079, A108019, A353158.

a(n) = { my (high=2^#binary(n), low=high-1-n); min(n-low, high-n) }

a(n) = min(A006257(n), A080079(n)) for any n > 0.
a(n) = 1 iff n belongs to A097110.
a(n) = n/2 iff n belongs to A020988.
a(n) = n/4 iff n belongs to A108019.
2*a(n) - a(2*n) = 0 or 1.

A352501 a(n) is the distance from n to the nearest integer that can be added to n without carries in balanced ternary.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 4, 4, 5, 4, 4, 2, 1, 1, 2, 4, 4, 5, 7, 10, 11, 13, 10, 11, 13, 13, 14, 13, 13, 11, 10, 13, 11, 10, 7, 5, 4, 4, 2, 1, 1, 2, 4, 4, 5, 7, 10, 11, 13, 10, 11, 13, 13, 14, 16, 19, 20, 22, 28, 29, 31, 31, 32, 34, 37, 38, 40, 28, 29, 31, 31, 32

Offset: 0

Rémy Sigrist, Apr 28 2022

Two integers can be added without carries in balanced ternary if they have no equal nonzero digit at the same position.

			For n = 7:
- the numbers k around 7, alongside their distance to 7, balanced ternary expansion and whether they require carries when added to 7, are:
      k   d  bter(k)  carries?
      --  -  -------  --------
       3  4       10  no
       4  3       11  yes
       5  2      1TT  yes
       6  1      1T0  yes
       7  0      1T1  yes
       8  1      10T  yes
       9  2      100  yes
      10  3      101  yes
      11  4      11T  yes
- so a(7) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..9841
Wikipedia, Balanced ternary

Cf. A003462, A007051, A059095, A353158.

ok(u,v) = { while (u && v, my (uu=[0,+1,-1][1+u%3], vv=[0,+1,-1][1+v%3]); if (abs(uu+vv)>1, return (0)); u=(u-uu)/3; v=(v-vv)/3); return (1) }
a(n) = for (d=0, oo, if (ok(n, n-d) || ok(n, n+d), return (d)))

a(n) = 1 iff n > 0 and n belongs to A003462 or A007051.

a(3*n) = 3*a(n)+1 for any n > 0.

cp's OEIS Frontend

A353157 a(n) is the distance from n to the nearest integer whose binary expansion has no common 1-bits with that of n.

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