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

1373, 1374, 1375, 1376, 1377, 1378, 1379, 1382, 1383, 1384, 1385, 1386, 1387, 1388, 1389, 1391, 1392, 1393, 1394, 1395, 1396, 1397, 1398, 1399, 1591, 1592, 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

Offset: 1

Author

Scott R. Shannon, Nov 24 2019

Keywords

• nonn
• base

Comments

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.

There are 466 terms below 10000.

From M. F. Hasler, Dec 02 2019: (Start)

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.

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.

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.)

(End)

Examples

			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.
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
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..13047. (All terms <= 50000.)

Crossrefs

Cf. A329623, A329624, A329197, A329200, A328865.

Programs

PARI

is_A329917(n, L=n^10, U=[n])=!for(i=1, oo, setsearch(U, n=A329623(n))&&return; nM. F. Hasler, Dec 02 2019

Extensions

Name and comment edited by M. F. Hasler, Dec 02 2019