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

Author

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

Keywords

• sign
• base

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

Examples

			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
  .....
		

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.)

Crossrefs

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

Programs

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 *)

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

Extensions

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