|
%I #59 Jan 23 2021 05:31:31
%S 0,1,1,1,1,1,1,1,1,1,3,48,14,76793,385,12,232,98,24,4,6,10952,8,
%T 575702095,18,82,39,16,7,6,26,9,8,30,12,13,182,449,25,62
%N 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.
%C 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.)
%C 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.
%C If a negative number appears in column a, say -107, then in column b we write successively -1, 0, -7.
%C 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
%C a(40) (at least 10^15) is the smallest unknown value.
%H Eric Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/MissNumbers.htm">Miss Numbers</a>, Feb 2012
%H Eric Angelini, <a href="/A208059/a208059.pdf">Miss Numbers</a> [Cached copy, with permission]
%H Christopher Creutzig, <a href="http://chesswanks.com/num/a208059/analogues.txt">A208059 analogues (bases 2-16)</a>
%H Christopher Creutzig, <a href="/A208059/a208059.txt">A208059 analogues (bases 2-16)</a> [Cached copy made Jun 01 2012]
%H Hans Havermann, <a href="http://chesswanks.com/num/a208059/">A208059 evolutions</a>
%e 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:
%e .a.....b......c
%e 12 - 1 = 11
%e 11 - 2 = 9
%e 9 - 1 = 8
%e 8 - 1 = 7
%e 7 - 9 = -2
%e -2 - 8 = -10
%e -10 - 7 = -17
%e -17 -(-2) = -15
%e -15 -(-1) = -14
%e -14 -(-0) = -14
%e -14 -(-1) = -13
%e -13 -(-7) = -6
%e -6 -(-1) = -5
%e -5 -(-5) = 0
%e etc., ad infinitum.
%e We get column b by reading column a digit-by-digit.
%t f[n_] := Module[{x = n, l, c}, c = 0; l = IntegerDigits[x];
%t While[x != 0, c++; x = x - First[l];
%t l = Join[Rest[l], Sign[x]*IntegerDigits[x]]; ]; c] ;
%t Table[f[n], {n, 0, 22}] (* _Robert Price_, Apr 04 2020 *)
%o (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
%Y Cf. A207505, A207506.
%K nonn,base,more
%O 0,11
%A _Hans Havermann_, Feb 23 2012
%E More terms from _Hans Havermann_, Mar 06 2012
%E Edited by _N. J. A. Sloane_, Jun 01 2012
|
