A237577 Let x(1)x(2)...x(q) denote the decimal expansion of a number n > 9. a(n) is the number of iterations needed to reach the end of the cycle under the map n -> f(n) where f(n) is represented by the number y(1)y(2)...y(q) such that y(k) = (x(k) + x(k+1)) mod 10 for k = 1..q-1 and y(q) = (x(q) + x(1)) mod 10.

5, 4, 5, 4, 2, 4, 5, 4, 5, 1, 4, 5, 3, 2, 4, 5, 4, 5, 1, 5, 5, 4, 2, 4, 5, 4, 5, 1, 5, 4, 4, 2, 4, 5, 3, 5, 1, 5, 4, 5, 2, 4, 5, 4, 5, 1, 5, 4, 5, 4, 4, 5, 4, 5, 1, 5, 3, 5, 4, 2, 5, 4, 5, 1, 5, 4, 5, 4, 2, 4, 4, 5, 1, 5, 4, 5, 4, 2, 3, 5, 5, 1, 5, 4, 5, 4, 2

Offset

10,1

Comments

The smallest integers such that the number of iterations equals 1, 2, 3, ... are 19, 14, 22, 11, 10, 104, 106, 110, 119, 108, 112, 101, 100, 114, 121, 502, 1041, 1402, 10110, 10048, 10010, 10202, 10113, 10116, 10118, 10026, 11294, 10281, 10282, 10120, ...

Links

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

Example

a(10) = 5 because 10 -> 11 -> 22 -> 44 -> 88 -> 66 with 5 iterations where 66 is the last number of the cycle because 66 -> 22, which has already appeared in the trajectory.
a(327) = 9 because 327 -> 590 -> 495 -> 349 -> 732 -> 59 -> 44 -> 88 -> 66 -> 22 with 9 iterations where 22 is the last number of the cycle because 22 -> 44, which has already appeared in the trajectory.

Maple

with(numtheory):nn:=100:for n from 10 to 200 do:lst:={n}:nn:=n:s:=1:for it from 1 to 40 while (s<>0)do: x:=convert(nn, base, 10):n1:=nops(x):s:=sum('irem(x[i]+x[i+1], 10)*10^i', 'i'=1..n1-1):s:=s+irem(x[n1]+x[1], 10):lst:=lst union {s}:nn:=s:od:n2:=nops(lst)-1: printf(`%d, `, n2):od:

Crossrefs

Sequence in context: A305394 A361803 A344124 * A131369 A122219 A093348

Adjacent sequences: A237574 A237575 A237576 * A237578 A237579 A237580

Keyword

nonn,base

Author

Michel Lagneau, Feb 09 2014

Extensions

Name edited by Jon E. Schoenfield, Sep 12 2017

Status

approved