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

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

Author

Rémy Sigrist, Apr 28 2022

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A003462, A007051, A059095, A353158.

Programs

PARI

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)))

Formula

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

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