A178049 The number of iterations of the map x -> x + d(x) to reach a repdigit (cf. A010785), starting at n, where the decimals of d(x) are the first differences of the decimals of x.

2, 1, 7, 6, 7, 5, 3, 6, 8, 4, 2, 2, 1, 5, 4, 7, 3, 3, 6, 8, 2, 2, 2, 1, 5, 4, 7, 3, 3, 6, 2, 2, 2, 2, 1, 5, 4, 3, 3, 3, 2, 2, 2, 2, 2, 1, 5, 4, 3, 3, 2, 2, 2, 2, 2, 2, 1, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 10

Author

Michel Lagneau, May 18 2010

Keywords

• nonn
• base

Comments

Let the recurrence X(k+1) = X(k) + Y(k) with the initial values : X(0) = n, and {x(1), x(2),...,x(p) } is the decimal expansion of n ; Y(0) has the decimal expansion {y(2), y(3),...,y(p)} where y(i) = abs(x(i)- x(i-1)), i = 2,..., p. For n > = 10, a(n) is the number of iterations of Y(k) needed to reach 0.

According to the computations with the Maple program for big numbers, the recurrence converges. Y(k) tends towards zero after a number of finite iterations, and X(k) tends towards a number q with the decimal expansion {p,p, ...,p }.

Examples

			a(11) = 1 because 11 + (1-1) = 11 + 0, and 0 is obtained after the first iteration.
a(12) = 7 because 12 + 1 = 13 -> 13 + 2 = 15 -> 15 + 4= 19 -> 19 + 8 = 27-> 27 + 5 = 32 -> 32 + 1 = 33 -> 33 + 0 = 33 is the last number of the cycle, and 0 is obtained after the 7th iteration.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 10..10000

Crossrefs

Cf. A010785.

Programs

Maple

for n from 10 to 200 do:n0:=n:s:=1:for i from 1 to 10^6 while(s<>0) do:x:=convert(n0,base,10):n1:=nops(x):s:=sum(abs(x[j+1]-x[j]),j=1..n1-1):n0:=n0+s:it:=i:od: printf(`%d, `, it):od: