0,1

Although this entry was only created in January, 2023, the problem had already been extensively studied in 2022.

Comment from Michael S. Branicky, Jul 26 2022: (Start)

Starting at 11, this first reaches 0 at step 1399141.

The longest string encountered has length 222:

444444414144444454144444145454455155154545515454564756517555545657676664\

465677675961617616416561527541551562575592651853254255356658359962263264\

365667368971272273374676377982812823836853869892911922935952968991101010\

121016.

(End)

It is conjectured that every starting number will eventually enter a cycle or reach 0 (see A359142 for small examples).

The first nontrivial cycle has length 583792 and the smallest number in it is 3374 (see the "Cycles in ..." Havermann link).

There is no b-file, but instead there is an a-file from Hans Havermann giving the sequence in full in b-file format, from a(0) to a(1399141). Beware, this is a 106.5 MB file. - N. J. A. Sloane, Feb 01 2023

From Michael S. Branicky, Sep 06 2023: (Start)

There are additional cycles with lengths

- 20173, containing 34674044445,

- 46, containing 9982228989928229222222829202026260298265278295291026. (End)

Eric Angelini, Does this iteration end? (Sum and erase), Personal blog "Cinquante Signes", blogspot.com, Jul 26 2022.

Eric Angelini, Does this iteration end? (Sum and erase), Personal blog "Cinquante Signes", blogspot.com, Jul 26 2022. [Cached copy, pdf file, with permission]

Michael S. Branicky, Python program for the sum-and-erase sequence

Jean-Paul Delahaye, Des suites à la dynamique insaisissable, Pour la Science #549, July 2023, pp. 80-85. (Link requires a subscription.)

Hans Havermann, Table of n, a(n) for n = 0..1399141 [Beware, this is a 106.5 MB file.]

Hans Havermann, Cycles in Éric Angelini's sum-and-erase, Glad Hobo Express Blog, Jul 28 2022

Hans Havermann, A cycle of length 49 [Astonishing! - N. J. A. Sloane, Jan 31 2023]

N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)

a[1] = {1, 1}; nn = 21; Do[If[FreeQ[#3, #2], Set[k, #1~Join~#3], Set[k, #1~Join~#3]; Set[k, DeleteCases[#1~Join~#3, #2]]] & @@ {#, First[#], IntegerDigits@ Total[#]} &[a[n - 1]]; Set[a[n], k], {n, 2, nn}]; Array[(1 - 2 Boole[First[#] == 0])*FromDigits@ # &@ a[#] &, nn] (* Michael De Vlieger, Mar 16 2023 *)

Cf. A359142, A361350, A361501.

Sequence in context: A132926 A094704 A019523 * A375101 A361350 A361501

Adjacent sequences: A359140 A359141 A359142 * A359144 A359145 A359146

sign,base

N. J. A. Sloane, Jan 31 2023, based on suggestions from Eric Angelini and Hans Havermann

approved