A368940 Number of iterations before a repeated value, or -1 if this never occurs, when starting at k = 1 and repeating k = k*n if k does not contain any adjacent equal digits, else k = k with all adjacent equal digits replaced by a single copy of the same digit.

1, 86, 338, 816, 2031, 1570, 2637, 2392, 4790, 3, 2, 15199, 21136, 8124, 12360, 18210, 101998, 41798, 15250, 135, 40063, 27298, 176470, 6553, 15757, 5031, 187645, 24050, 567055, 487, 141008, 71243, 341907, 154758, 38175, 150429, 84011, 106833, 351884, 1117, 391266, 324631, 1287699, 374743

Offset: 1

Author

Scott R. Shannon and Eric Angelini, Jan 10 2024

Keywords

• nonn
• base

Comments

The largest term in the first 140 terms is a(127) = 121311726, which reaches a maximum value of 133672219006681613318653118648140533992241 at the 17276871st iteration, before repeating 2102014745703.

Examples

			a(2) = 86 as the iterations are : 1 -> 2 -> 4 -> 8 -> 16 -> 32 -> 64 -> 128 -> 256 -> 512 -> 1024 -> 2048 -> 4096 -> 8192 -> 16384 -> 32768 -> 65536 -> 6536 -> 13072 -> 26144 -> 2614 -> 5228 -> 528 -> 1056 -> 2112 -> 212 -> 424 -> 848 -> 1696 -> 3392 -> 392 -> 784 -> 1568 -> 3136 -> 6272 -> 12544 -> 1254 -> 2508 -> 5016 -> 10032 -> 1032 -> 2064 -> 4128 -> 8256 -> 16512 -> 33024 -> 3024 -> 6048 -> 12096 -> 24192 -> 48384 -> 96768 -> 193536 -> 387072 -> 774144 -> 7414 -> 14828 -> 29656 -> 59312 -> 118624 -> 18624 -> 37248 -> 74496 -> 7496 -> 14992 -> 1492 -> 2984 -> 5968 -> 11936 -> 1936 -> 3872 -> 7744 -> 74 -> 148 -> 296 -> 592 -> 1184 -> 184 -> 368 -> 736 -> 1472 -> 2944 -> 294 -> 588 -> 58 -> 116 -> 16, taking 86 steps to reach a repeated value.
a(10) = 3 as the iterations are : 1 -> 10 -> 100 -> 10, taking three steps to reach a repeated value.
a(11) = 2 as the iterations are : 1 -> 11 -> 1, taking two steps to reach a repeated value.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..220 (terms 1..140 from Scott R. Shannon)
• Eric Angelini, Replace my twins, triplets, etc. by 1, personal blog CinquanteSignes.blogspot.com, Jan. 5, 2024.

Crossrefs

Cf. A171901, A351328, A330674, A316650, A098282, A330634, A329624.

Programs

Python

from itertools import groupby
def a(n):
    seen, k, c = set(), 1, 0
    while k not in seen:
        seen.add(k)
        c += 1
        s = str(k)
        t = "".join(k for k, g in groupby(s))
        k = k*n if s == t else int(t)
    return c
print([a(n) for n in range(1, 45)]) # Michael S. Branicky, Jan 11 2024