login
For any n >= 0, let x_n(1) = n, and for any b > 1, x_n(b) = x_n(b-1) minus the sum of digits of x_n(b-1) in base b; a(n) is the least b such that x_n(b) = 0.
3

%I #13 Aug 07 2022 15:36:45

%S 1,2,3,3,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,8,8,8,8,8,

%T 8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,11,11,11,

%U 11,11,11,12,12,12,12,12,12,12,12,13,13,13,13,13

%N For any n >= 0, let x_n(1) = n, and for any b > 1, x_n(b) = x_n(b-1) minus the sum of digits of x_n(b-1) in base b; a(n) is the least b such that x_n(b) = 0.

%C This sequence is well defined: for any n >= 0: if x_n(b) > 0, then x_n(b+1) < x_n(b), and we must eventually reach 0.

%C This sequence is weakly increasing; this is related to the fact that for any base b > 1, k -> (k minus the sum of digits of k in base b) is weakly increasing.

%C Note that some values (like 7) do not appear in this sequence (see also A356386).

%H Rémy Sigrist, <a href="/A356384/b356384.txt">Table of n, a(n) for n = 0..10000</a>

%H Rémy Sigrist, <a href="/A356384/a356384.png">Colored scatterplot of (n, x_n(b)) for n <= 1000 and b = 1..a(n)</a> (the color is function of b)

%H Rémy Sigrist, <a href="/A356384/a356384.gp.txt">PARI program</a>

%e For n = 42:

%e - we have:

%e b x(b)

%e - ----

%e 1 42

%e 2 39

%e 3 36

%e 4 33

%e 5 28

%e 6 20

%e 7 12

%e 8 7

%e 9 0

%e - so a(42) = 9.

%o (PARI) See Links section.

%Y Cf. A011371, A066568, A071542, A261231, A344853, A356386.

%K nonn,base

%O 0,2

%A _Rémy Sigrist_, Aug 05 2022