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A321882 a(n) is the least base b > 1 such that the sum n + n can be computed without carry. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 6 12:08 EDT 2020. Contains 333273 sequences. (Running on oeis4.)