

A208059


Start with n, successively subtract each digit of the resulting sequence (the digits of a negative term being the negatives of that term's digits): a(n) is the number of steps needed to get to the first zero.


16



0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 48, 14, 76793, 385, 12, 232, 98, 24, 4, 6, 10952, 8, 575702095, 18, 82, 39, 16, 7, 6, 26, 9, 8, 30, 12, 13, 182, 449, 25, 62
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OFFSET

0,11


COMMENTS

This is the same procedure used in A207505 with an allowance made to continue the process if we miss zero and enter negative territory on our initial downward run. A downward run will succumb to an upward run, and vice versa, with each run presenting another opportunity to hit zero. (A randomdigit trail entails, per run, a 1 in 5 chance of hitting zero.)
a(23) was first computed by Nicolas Berr. According to his calculations, a(40) does not hit zero in its first 15 signchange crossings. The 16th crossing is ~2*10^15.
If a negative number appears in column a, say 107, then in column b we write successively 1, 0, 7.
As far as I know, it is only a conjecture that for any starting value n we always reach a 0.  N. J. A. Sloane, Jun 01 2012
a(40) (at least 10^15) is the smallest unknown value.


LINKS

Table of n, a(n) for n=0..39.
Eric Angelini, Miss Numbers, Feb 2012
Eric Angelini, Miss Numbers [Cached copy, with permission]
Christopher Creutzig, A208059 analogues (bases 216)
Christopher Creutzig, A208059 analogues (bases 216) [Cached copy made Jun 01 2012]
Hans Havermann, A208059 evolutions


EXAMPLE

When successively subtracting its own digittrail, 12 requires 14 steps to hit its first zero, achieved on its first upward run, thus making a(12) = 14:
.a.....b......c
12  1 = 11
11  2 = 9
9  1 = 8
8  1 = 7
7  9 = 2
2  8 = 10
10  7 = 17
17 (2) = 15
15 (1) = 14
14 (0) = 14
14 (1) = 13
13 (7) = 6
6 (1) = 5
5 (5) = 0
etc., ad infinitum.
We get column b by reading column a digitbydigit.


MATHEMATICA

f[n_] := Module[{x = n, l, c}, c = 0; l = IntegerDigits[x];
While[x != 0, c++; x = x  First[l];
l = Join[Rest[l], Sign[x]*IntegerDigits[x]]; ]; c] ;
Table[f[n], {n, 0, 22}] (* Robert Price, Apr 04 2020 *)


PROG

(PARI) A208059(n, v=0/*verbose: print all terms if >0*/, a=[])={ v&print1(n); a=eval(Vec(Str(n))); for(c=0, 9e9, nreturn(c); a=concat(vecextract(a, "^1"), eval(Vec(Str(abs(n=a[1]))))*sign(n)); v&print1(", "n)) } \\ M. F. Hasler, Mar 03 2012


CROSSREFS

Cf. A207505, A207506.
Sequence in context: A088718 A219162 A016548 * A162333 A003878 A219056
Adjacent sequences: A208056 A208057 A208058 * A208060 A208061 A208062


KEYWORD

nonn,base,more


AUTHOR

Hans Havermann, Feb 23 2012


EXTENSIONS

More terms from Hans Havermann, Mar 06 2012
Edited by N. J. A. Sloane, Jun 01 2012


STATUS

approved



