login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A328975 Numbers whose trajectory under repeated application of the map in A053392 increases without limit. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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 staring 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.

Sequence in context: A067841 A237968 A062912 * A328974 A236732 A184078

Adjacent sequences:  A328972 A328973 A328974 * A328976 A328977 A328978

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Nov 02 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 4 02:18 EST 2020. Contains 338921 sequences. (Running on oeis4.)