A323832 - OEIS

%I #38 Feb 13 2019 17:24:17

%S 0,0,19,12,18,0,11,23,17,4,19,1,10,29,22,32,16,5,3,47,18,15,1,20,9,2,

%T 28,26,21,13,31,24,15,1,4,23,2,18,46,21,17,51,14,15,1,24,19,2,8,10,1,

%U 33,27,24,25,1,20,19,12,18,30,1,23,7,14,29,5,20,3

%N Start with n and repeatedly double it and apply the "repeatedly delete any run of identical digits" operation described in A323830; a(n) is the number of steps needed to reach one of 0, 1, or 5, or -1 if none of these three numbers is ever reached.

%C Conjecture: every number will eventually reach one of 0, 1, or 5.

%C From _Chai Wah Wu_, Feb 04 2019, Feb 13 2019: (Start)

%C Conjecture is true for n < 10^10.

%C 1604466 takes 115 steps to reach 5 and is the largest value for a(n) for n < 10^7.

%C 91070713 takes 121 steps to reach 5 and is the largest value for a(n) for n < 10^8.

%C 126591463 and 801282051 both take 128 steps to reach 5 and this is the largest value for a(n) for n < 10^9.

%C The numbers 1582393271, 1582393293, 4645106705 all take 131 steps to reach 5 and this is the largest value for a(n) for n < 10^10.

%C (End)

%H Chai Wah Wu, <a href="/A323832/b323832.txt">Table of n, a(n) for n = 0..10000</a>

%e Starting with 2, the trajectory is 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 636, 1272, 25, 50, 1, reaching 1 in 20 steps, so a(2) = 20.

%e 3 reaches 1 in 12 steps: 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 61, 1, so a(3) = 12.

%e 10 reaches 5 in 19 steps: 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 6360, 12720, 250, 5, so a(10) = 19.

%o (Python)

%o from re import split

%o def A321801(n):

%o     return int('0'+''.join(d for d in split('(0+)|(1+)|(2+)|(3+)|(4+)|(5+)|(6+)|(7+)|(8+)|(9+)',str(n)) if d != '' and d != None and len(d) == 1))

%o def f(n):

%o     x = 2*n

%o     y = A321801(x)

%o     while x != y:

%o         x, y = y, A321801(y)

%o     return x

%o def A323832(n):

%o     mset, m, c = set(), n, 0

%o     while True:

%o         if m == 1 or m == 0 or m == 5:

%o             return c

%o         m = f(m)

%o         if m in mset:

%o             return -1

%o         mset.add(m)

%o         c += 1 # _Chai Wah Wu_, Feb 04 2019, Feb 11 2019

%Y Cf. A321801, A321802, A323830, A323830.

%K nonn,base

%O 0,3

%A _N. J. A. Sloane_, Feb 03 2019