A321909 a(n) is the least base b > 1 in which the additive persistence of n is <= 1.

2, 2, 2, 3, 2, 4, 3, 5, 2, 3, 3, 5, 3, 6, 6, 5, 2, 4, 3, 6, 4, 4, 7, 7, 4, 5, 5, 3, 3, 7, 3, 5, 2, 4, 8, 5, 3, 6, 6, 6, 5, 8, 6, 6, 6, 6, 9, 9, 4, 6, 5, 5, 5, 7, 3, 5, 5, 7, 7, 7, 5, 10, 10, 7, 2, 4, 4, 8, 4, 4, 7, 7, 4, 6, 6, 5, 5, 7, 6, 6, 4, 3, 3, 8, 3, 6

Offset: 0

Author

Rémy Sigrist, Nov 21 2018

Keywords

• nonn
• base

Comments

Equivalently, a(n) is the least base b > 1 in which the sum of digits of n is < b.

The sequence is well defined as, for any n > 0, the additive persistence of n is 0 in base n + 1.

This sequence is unbounded.

Examples

			For n = 42:
- in base 2, 42 has additive persistence 3: "101010" -> "11" -> "10" -> "1",
- in base 3, 42 has additive persistence 2: "1120" -> "11" -> "2",
- in base 4, 42 has additive persistence 2: "222" -> "12" -> "3",
- in base 5, 42 has additive persistence 2: "132" -> "11" -> "2",
- in base 6, 42 has additive persistence 1: "110" -> "2",
- 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..1000000 (where the color is function of the initial digit of n in base a(n))

Crossrefs

See A321882 for a similar sequence.

Cf. A031286, A131577.

Programs

Mathematica

Array[Block[{b = 2}, While[Total@ IntegerDigits[#, b] >= b, b++]; b] &, 86, 0] (* Michael De Vlieger, Nov 25 2018 *)

PARI

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

Formula

a(n) = 2 iff n belongs to A131577.

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