A329200 The ghost iteration (A): add or subtract the number formed by absolute differences of digits (A040115), according to parity (even or odd).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 11, 11, 15, 11, 19, 11, 23, 11, 27, 22, 20, 22, 22, 26, 22, 30, 22, 34, 22, 27, 33, 31, 33, 33, 37, 33, 41, 33, 45, 44, 38, 44, 42, 44, 44, 48, 44, 52, 44, 45, 55, 49, 55, 53, 55, 55, 59, 55, 63, 66, 56, 66, 60, 66, 64, 66, 66, 70, 66, 63, 77, 67, 77, 71, 77

Offset

0,3

Comments

Sequence A040115 is most naturally extended to 0 (empty sum) for single-digit arguments; that's what we use here. This value is added to n if even, subtracted if odd.

Repdigit numbers are the fixed points. Other starting values end in nontrivial loops under iterations of this map, like 11090 -> 10891 -> 12709 -> 11130 -> 11107 -> 11090 etc. Table A329196 lists these cycles, A329197 their lengths.

A329198 gives the size of n's orbit, i.e., the length of the trajectory until the terminating cycle is covered.

Links

Table of n, a(n) for n=0..75.

E. Angelini, The ghost iteration, SeqFan list, Nov 2019

E. Angelini, The Ghost Iteration, Personal blog "Cinquante signes", Nov 2019

Formula

a(n) = n + (-1)^d*d where d = A040115(n), 0 for n < 10.

Example

For n = 101, the number formed by the absolute differences of digits is 11, since this is odd it is subtracted from n, so a(101) = 101-11 = 90.

Prog

(PARI) apply( A329200(n)={n+(-1)^(n=fromdigits(abs((n=digits(n+!n))[^-1]-n[^1])))*n}, [1..199])

Crossrefs

Cf. A040115, A329201 (variant B: add/subtract if odd/even).

Cf. A329196 (cycles), A329197 (lengths), A329198 (size of orbit of n).

Sequence in context: A261914 A261423 A262040 * A122638 A297235 A090175

Adjacent sequences: A329197 A329198 A329199 * A329201 A329202 A329203

Keyword

nonn,base,easy

Author

Eric Angelini and M. F. Hasler, Nov 09 2019

Status

approved