A209674 For each n, define a sequence of numbers by S(0)=n, S(i) = sum of last two digits of the concatenation S(0)S(1)S(2)...S(i-1); a(n) = smallest m such that S(m) = 5, or -1 if 5 is never reached.

-1, 4, 9, 9, 5, 0, 4, 3, 10, 11, 5, 3, 2, 6, 1, 5, 8, 7, 9, 6, 10, 7, 8, 1, 7, 4, 3, 10, 6, 4, 10, 2, 1, 8, 5, 8, 7, 6, 4, 3, 6, 1, 4, 7, 4, 3, 6, 4, 3, 7, 1, 9, 11, 5, 8, 6, 4, 3, 7, 2, 5, 8, 7, 4, 6, 4, 3, 7, 2, 6, 4, 10, 5, 6, 4, 3, 7, 2, 6, 9, 11, 7, 6, 4, 3, 7, 2, 6, 9, 8, 12, 6, 4, 3, 7, 2, 6, 9, 8, 10

Offset: 0

Author

N. J. A. Sloane, Mar 11 2012

Keywords

• sign
• base

Comments

a(n) = -1 iff n ends in 00 (e.g. 100, 200, ...). (It is sufficient to check the 100 starts i,j, 0 <= i, j <= 9.)

5 is the unique number common to the trajectories of all numbers from 1 to 99.

Iterate the map k -> A209685(k), starting at n, until reaching 5, or -1 if 5 is never reached.

Examples

			For n=4 we have S(0)=4, S(1)=4, S(2)=8, S(3)=12, S(4)=3, S(5)=5, so a(4)=5.
		

References

• Eric Angelini, Posting to Math Fun Mailing List, Mar 11 2012.

Crossrefs

Cf. A209685, A209686.

Programs

PARI

a(n)=my(m=0,t,k);while(n!=5,t=if(n>9,n%100\10+n%10,n+m%10);m=n;n=t;k++);k \\ Charles R Greathouse IV

Formula

The sequence is ultimately periodic.

Extensions

Corrected and extended by Charles R Greathouse IV, Mar 11 2012