A359143 The sum-and-erase sequence starting at 11: a(0) = 11; for n>=1, let m = a(n-1), and if m < 0, change m to an improper decimal "number" by replacing the minus sign by a single leading zero; then a(n) = A359142(m).

11, 112, 1124, 11248, 2486, 4860, 486018, 48601827, 4860182736, 8601827365, 860182736546, 86018273654656, 8601827365465667, 601273654656670, -1273545704, -127354570438, -12735457043849, -1273545704384962, 1273545743849627, 127354574384962777, 273545743849627779

Offset

0,1

Comments

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)

Links

Table of n, a(n) for n=0..20.

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

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.)

Mathematica

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 *)

Crossrefs

Cf. A359142, A361350, A361501.

Sequence in context: A132926 A094704 A019523 * A375101 A361350 A361501

Adjacent sequences: A359140 A359141 A359142 * A359144 A359145 A359146

Keyword

sign,base

Author

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

Status

approved