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

0,1

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.

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

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.

Example

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.

Mathematica

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

Prog

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

Crossrefs

See A319478 for the multiplicative variant.

Cf. A005836, A007091.

Sequence in context: A072751 A251542 A131971 * A281158 A100742 A001269

Adjacent sequences: A321879 A321880 A321881 * A321883 A321884 A321885

Keyword

nonn,base

Author

Rémy Sigrist, Nov 20 2018

Status

approved