A321882 a(n) is the least base b > 1 such that the sum n + n can be computed without carry.

2, 3, 5, 3, 3, 4, 5, 5, 6, 3, 3, 5, 3, 3, 6, 7, 4, 4, 8, 8, 4, 4, 7, 7, 7, 5, 5, 3, 3, 9, 3, 3, 5, 10, 10, 5, 3, 3, 6, 3, 3, 10, 6, 6, 6, 11, 11, 11, 6, 6, 5, 5, 5, 12, 13, 5, 5, 5, 7, 7, 5, 5, 5, 7, 4, 4, 7, 8, 4, 4, 7, 7, 6, 6, 6, 8, 14, 15, 6, 6, 4, 3, 3, 8

Offset: 0

Author

Rémy Sigrist, Nov 20 2018

Keywords

• nonn
• base

Comments

Equivalently, a(n) is the least base b > 1 where:

- twice the greatest digit of n is < b,

- twice the digital sum of n equals the digital sum of twice n.

The sequence is well defined as, for any n > 0, n + n can be computed without carry in base 2*n + 1.

The sequence is unbounded; by contradiction:

- suppose that v = a(n) is the greatest term of the sequence,

- we can assume that v > 2,

- let d be the greatest digit of v!^A000120(n) in base v,

- let k = floor((v-1) / d),

- necessarily a(n + k * (v!^A000120(n))) > v, QED.

Examples

			For n = 42:
- in base 2, 42 + 42 cannot be computed without carry: "101010" + "101010" = "1010100",
- in base 3, 42 + 42 cannot be computed without carry: "1120" + "1120" = "10010",
- in base 4, 42 + 42 cannot be computed without carry: "222" + "222" = "1110",
- in base 5, 42 + 42 cannot be computed without carry: "132" + "132" = "314",
- in base 6, 42 + 42 can be computed without carry: "110" + "110" = "220",
- hence a(42) = 6.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 0..10000000 (where the color is function of the initial digit of n in base a(n))
• Index entries for sequences related to carryless arithmetic

Crossrefs

See A319478 for the multiplicative variant.

Cf. A005836, A007091.

Programs

Mathematica

Array[Block[{b = 2}, While[2 Max@ IntegerDigits[#, b] >= b, b++]; b] &, 84, 0] (* Michael De Vlieger, Nov 25 2018 *)

PARI

a(n) = for (b=2, oo, if (2*sumdigits(n, b)==sumdigits(n*2, b), return (b)))

Formula

a(n) = 2 iff n = 0.

a(n) = 3 iff n > 0 and n belongs to A005836.

a(n * a(n)) <= a(n).

a(A007091(n)) <= 10 for any n >= 0.