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A208059
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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.
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16
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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
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0,11
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COMMENTS
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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 random-digit 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 sign-change 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.
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LINKS
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EXAMPLE
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When successively subtracting its own digit-trail, 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 digit-by-digit.
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MATHEMATICA
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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] ;
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PROG
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(PARI) A208059(n, v=0/*verbose: print all terms if >0*/, a=[])={ v&print1(n); a=eval(Vec(Str(n))); for(c=0, 9e9, n|return(c); a=concat(vecextract(a, "^1"), eval(Vec(Str(abs(n-=a[1]))))*sign(n)); v&print1(", "n)) } \\ M. F. Hasler, Mar 03 2012
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CROSSREFS
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KEYWORD
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nonn,base,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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