A329201 The ghost iteration (B): add or subtract the number formed by absolute differences of digits (A040115), according to parity (odd or even).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 13, 11, 17, 11, 21, 11, 25, 11, 18, 22, 22, 24, 22, 28, 22, 32, 22, 36, 33, 29, 33, 33, 35, 33, 39, 33, 43, 33, 36, 44, 40, 44, 44, 46, 44, 50, 44, 54, 55, 47, 55, 51, 55, 55, 57, 55, 61, 55, 54, 66, 58, 66, 62, 66, 66, 68, 66, 72, 77, 65, 77, 69, 77, 73, 77

Offset: 0

Author

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

Keywords

• nonn
• base
• easy

Comments

Sequence A040115 is most naturally extended to 0 (empty sum) for single-digit arguments; that's what we use for n < 10 here. This value is subtracted from n if even, added if odd.

A040115 is zero iff the argument is a repdigit (A010785), which therefore are the fixed points of this map A329201. All small starting values reach a fixed point, but larger values may enter a nontrivial cycle (or "loop").

See the table A329342 for the list of these cycles.

Examples

			For n = 101, the number formed by the absolute differences of digits is 11. Since this is odd it is added to n, so a(101) = 101 + 11 = 112.
		

Links

• E. Angelini, The ghost iteration, SeqFan list, Nov 2019.
• E. Angelini, The ghost iteration, SeqFan list, Nov 2019 [Cached copy, with permission]
• E. Angelini, The Ghost Iteration, Personal blog "Cinquante signes", Nov 2019.
• E. Angelini, The Ghost Iteration, Personal blog "Cinquante signes", Nov 2019 [Cached copy, pdf file, with permission]

Crossrefs

Cf. A040115, A329200 (variant A: add/subtract if even/odd), A010785 (fixed points).

Cf. A329342 (list of cycles).

Programs

PARI

apply( A329201(n)={n-(-1)^(n=fromdigits(abs((n=digits(n+!n))[^-1]-n[^1])))*n}, [1..199])

Formula

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