A359142 Let s = sum of digits of n, let t = decimal concatenation of n and s, let u be obtained by deleting all copies of the leading digit of t from t, if this digit occurs in s. Then if u has only zero digits, a(n) = 0; if u has leading digit 0 but not all its digits are 0, delete all leading 0's from u and negate the result to get a(n); otherwise a(n) = u.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 112, 123, 134, 145, 156, 167, 178, 189, 90, 0, 213, 224, 235, 246, 257, 268, 279, 2810, 2911, 0, 314, 325, 336, 347, 358, 369, 3710, 3811, 3912, 0, 415, 426, 437, 448, 459, 4610, 4711, 4812, 4913, 0, 516, 527, 538, 549, 5510, 5611, 5712, 5813

Offset

1,11

Comments

Since nonzero numbers may not have leading zeros, we indicate their presence by negating the number.

For use in A359142, we also define a(n) for improper decimal numbers n with leading zeros by following exactly the same steps. To get a(073), for example, we append the digit sum 10, and delete the 0's from 07310, getting a(073) = 731. To get a(074), we append 11, getting 07411, so a(074) = -7411.

The sequence of n such that a(n) < 0 begins (109, 1009, 1018, 1019, 1027, 1028, 1029, 1036, ...): see A359144. - M. F. Hasler, Feb 01 2023

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

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]

Hans Havermann, Cycles in Eric Angelini's sum-and-erase, Personal blog "Glad Hobo Express", blogspot.com, Jul 28 2022.

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.)

Example

Examples:
  n.....s......t....u.....a(n)
  1.....1.....11....0......0
  2.....2.....22....0......0
  .....
  9.....9.....99....0......0
  10....1....101....0......0
  11....2....112..112....112
  12....3....123..123....123
  .....
  100...1...1001...00......0
  101...2...1012.1012...1012
  102...3...1023.1023...1023
  .....
  109..10..10910..090....-90
  .....

Mathematica

A359142[n_]:= Module[{d=IntegerDigits[n], s, u}, If[MemberQ[s=IntegerDigits[Total[d]], First[u=Join[d, s]]], u=DeleteCases[u, First[u]]]; If[u=={}||First[u]==0, -1, 1]FromDigits[u]]; Array[A359142, 100] (* Paolo Xausa, Oct 11 2023 *)

Prog

(PARI) A359142(n) = { my(d=digits(n), s=digits(vecsum(d))); n>0 || d=concat(0, d); n=concat(d, s); setsearch(Set(s), d[1]) && n=select(c->c!=d[1], n); if(n && !n[1], -fromdigits(n), fromdigits(n)) }
apply(A359142, [0..99]) \\ M. F. Hasler, Feb 01 2023
(Python)
def a(n):
    n = str(-n) if isinstance(n, int) and n < 0 else str(n)
    s = str(sum(map(int, n)))
    t = n + s
    u = t.replace(t[0], "") if t[0] in s else t
    return 0 if u == "" else (-int(u) if u[0] == "0" else int(u))
print([a(n) for n in range(1, 59)]) # Michael S. Branicky, Feb 01 2023

Crossrefs

Cf. A359143, A359144 (k such that a(k)<0).

Sequence in context: A281926 A036301 A117723 * A329719 A157662 A340125

Adjacent sequences: A359139 A359140 A359141 * A359143 A359144 A359145

Keyword

sign,base

Author

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

Extensions

More terms from M. F. Hasler, Feb 01 2023

Status

approved