A328975 Numbers whose trajectory under repeated application of the map in A053392 increases without limit.

1496, 1497, 1498, 1499, 1587, 1588, 1589, 1691, 1692, 1693, 1694, 1695, 1696, 1697, 1698, 1699, 1719, 1728, 1729, 1783, 1784, 1785, 1786, 1787, 1788, 1789, 1791, 1792, 1793, 1794, 1795, 1796, 1797, 1798, 1799, 1819, 1867, 1868, 1869, 1874, 1875, 1876, 1877, 1878, 1879, 1880, 1881, 1882, 1883, 1884, 1885

Offset: 1

Author

N. J. A. Sloane, Nov 02 2019

Keywords

• nonn
• base

Comments

Computed by Hans Havermann, Nov 01 2019.

There is a simple technique, due to Hans Havermann, that proves that many terms that in this sequence blow up: find a term in the trajectory that has an internal substring of three 9's. See A328974 for the case 1496.

The same reasoning shows that almost all numbers belong to the sequence.

From Scott R. Shannon, Nov 23 2019: (Start)

Although many of the numbers in this sequence eventually reach a term that contains three 9's, some do not. The first such example is 4949 which leads to 44444 after two steps, and starts a sequence of values that leads back to a much larger all-4's number every nine steps. No 9's appear in any of the intermediate values. Similar series found which lead to limitless loops are values with all 4's with a final 5, or all 3's with a final digit of 1 to 6. Of the first 33909 starting values that increase without limit 209 lead into one of these non-9 loops. (End)

From Scott R. Shannon, Nov 24 2019: (Start)

One can show that strings of repeated digits longer than a certain length will increase without limit by calculating the number of digits when a new value containing only the same digit appears again in the iterative sequence. Below shows the details of this for digits 1 to 9. The number of cycles is the number of iterations of A053392 required before a new value containing only the starting digit is seen. The minimum starting value is the smallest same-digit number such that subsequent iterations will produce another value with only the same digit of equal or longer length. The number of digits in the subsequence reoccurrence shows the number of digits in this value given the starting value has n digits. All sufficiently long single-digit numbers, with digits 1 to 8, increase 8-fold in length, minus a constant, after 9 iterations.

.

digit | # of cycles | min start value | # digits in reoccurrence

1 | 9 | 1111111 | 8*n - 45

2 | 9 | 222222 | 8*n - 38

3 | 3 | 33333 | 2*n - 5

4 | 9 | 44444 | 8*n - 31

5 | 9 | 55555 | 8*n - 33

6 | 3 | 6666 | 2*n - 4

7 | 9 | 77777 | 8*n - 27

8 | 9 | 8888 | 8*n - 28

9 | 2 | 999 | 2*n - 3

(End)

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..33909 (terms 1..109 from Hans Havermann).
• Hans Havermann, Proof of correctness of first 109 terms

Crossrefs

Cf. A053392, A328974.