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

%I A321909 #18 Aug 31 2021 02:43:43
%S A321909 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,
%T A321909 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,
%U A321909 8,4,4,7,7,4,6,6,5,5,7,6,6,4,3,3,8,3,6
%N A321909 a(n) is the least base b > 1 in which the additive persistence of n is <= 1.
%C A321909 Equivalently, a(n) is the least base b > 1 in which the sum of digits of n is < b.
%C A321909 The sequence is well defined as, for any n > 0, the additive persistence of n is 0 in base n + 1.
%C A321909 This sequence is unbounded.
%H A321909 Rémy Sigrist, <a href="/A321909/b321909.txt">Table of n, a(n) for n = 0..10000</a>
%H A321909 Rémy Sigrist, <a href="/A321909/a321909.png">Colored scatterplot of (n, a(n)) for n = 0..1000000</a> (where the color is function of the initial digit of n in base a(n))
%F A321909 a(n) = 2 iff n belongs to A131577.
%F A321909 a(n * a(n)) <= a(n).
%e A321909 For n = 42:
%e A321909 - in base 2, 42 has additive persistence 3: "101010" -> "11" -> "10" -> "1",
%e A321909 - in base 3, 42 has additive persistence 2: "1120" -> "11" -> "2",
%e A321909 - in base 4, 42 has additive persistence 2: "222" -> "12" -> "3",
%e A321909 - in base 5, 42 has additive persistence 2: "132" -> "11" -> "2",
%e A321909 - in base 6, 42 has additive persistence 1: "110" -> "2",
%e A321909 - hence a(42) = 6.
%t A321909 Array[Block[{b = 2}, While[Total@ IntegerDigits[#, b] >= b, b++]; b] &, 86, 0] (* _Michael De Vlieger_, Nov 25 2018 *)
%o A321909 (PARI) a(n) = for (b=2, oo, if (sumdigits(n, b) < b, return (b)))
%Y A321909 See A321882 for a similar sequence.
%Y A321909 Cf. A031286, A131577.
%K A321909 nonn,base
%O A321909 0,1
%A A321909 _Rémy Sigrist_, Nov 21 2018