cp's OEIS Frontend

A329917 Starting values for which iterations of A329623 diverge (conjectural).

%I A329917 #38 Dec 08 2019 08:11:45
%S A329917 1373,1374,1375,1376,1377,1378,1379,1382,1383,1384,1385,1386,1387,
%T A329917 1388,1389,1391,1392,1393,1394,1395,1396,1397,1398,1399,1591,1592,
%U A329917 1593,1594,1595,1596,1597,1598,1599,1891,1892,1893,1894,1895,1896,1897,1898,1899,2373,2374,2375,2376,2377,2378,2379,2382,2383,2384
%N A329917 Starting values for which iterations of A329623 diverge (conjectural).
%C A329917 These are the starting values n for which the trajectory under iterations of A329623 grows without limit. See A329624 for an explanation of the sequence.
%C A329917 There are 466 terms below 10000.
%C A329917 From _M. F. Hasler_, Dec 02 2019: (Start)
%C A329917 There is no term below 10^3, but beyond 10^4 they become much more frequent: roughly 1/3 of all numbers between 10^4 and 5*10^4 are in the sequence.
%C A329917 The sequence consists mostly in runs of consecutive numbers ending with the next larger term with final digit 9: 1373..1379, 1382..1389, 1391..1399, 1591..1599, 1891..1899, 2373..2379, ... In some ranges, like a(152) = 4010 to a(240) = 4181, a(307) = 6010 to a(352) = 6391, a(467) = 10010 to a(556) = 11090, ..., this pattern is reversed: the runs start with a multiple of 10.
%C A329917 However, all these terms are so far only conjectural. We have no proof that the terms for which the trajectory seems to "explode" do not eventually end up in a cycle. For example, the 8th iterate of a(1) = 1373 is 5218725017016262626273. If the 2nd digit is changed from 2 to 0, then the further iterates grow to a length of 52 digits, but finally end up in a 2-cycle of 45-digit numbers (26...26273, 62...62637). (All members of cycles are listed in A328142.)
%C A329917 (End)
%H A329917 Scott R. Shannon, <a href="/A329917/b329917.txt">Table of n, a(n) for n = 1..13047</a>. (All terms <= 50000.)
%e A329917 The first term to diverge is n = 1373. The iterative sequence begins 1373, 39637, 1176273, 26962637, 8124626273, 85486262637, 13826662626273, 411094294626262637, 5218725017016262626273, 68697250170162626262637, 141346472501701626262626273, ... The digits '62637' reappear at the end of the terms every second iteration.
%e A329917 While 50, 500 and 5000 reach the fixed point 9, 455, resp. 4444455 after 5, 3, resp. 8 iterations, the starting value 50000 is in this sequence: after the 10th iteration, the result is of the form 991...903544444455 and keeps this form (prefix alternating with 1810....) with an ever growing string of 4's. - _M. F. Hasler_, Dec 02 2019
%o A329917 (PARI) is_A329917(n, L=n^10, U=[n])=!for(i=1, oo, setsearch(U, n=A329623(n))&&return; n<L||break; U=setunion(U, [n])) \\ The experimental search limit n^10 could be replaced by something better. The starting value 1673 goes beyond 3*10^31 before reaching the cycle (26262626262626262626262626273, 62626262626262626262626262637). - _M. F. Hasler_, Dec 02 2019
%Y A329917 Cf. A329623, A329624, A329197, A329200, A328865.
%K A329917 nonn,base
%O A329917 1,1
%A A329917 _Scott R. Shannon_, Nov 24 2019
%E A329917 Name and comment edited by _M. F. Hasler_, Dec 02 2019