

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
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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((v1) / 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



