A392158 For each starting value n, the number of distinct terms in the sequence obtained by repeatedly doubling and then deleting repeated digits (keeping the first occurrence) until a term repeats.

30, 29, 35, 28, 31, 34, 37, 27, 40, 30, 30, 33, 13, 36, 36, 26, 36, 39, 39, 29, 42, 29, 35, 32, 32, 12, 32, 35, 28, 35, 38, 25, 35, 35, 38, 38, 37, 38, 48, 28, 12, 41, 38, 28, 41, 34, 41, 31, 51, 31, 21, 11, 44, 31, 31, 34, 37, 27, 40, 34, 34, 37, 34, 24, 10, 47, 44, 34

Offset

1,1

Comments

It will be very interesting to find the numbers that the sequence does not end in 104, for example 41 and 82, and also if there are other numbers that are the end of the sequences besides 104 and 164 (see examples).

From Michael S. Branicky, Jan 01 2026: (Start)

Computing A392158(k) for all 8877691 terms in A010784, the maximum is 68, achieved by:

380947615, 2840136957, 3129508674, 3145086729, 3401286957, 3419875206, 3809472615, 3809476215

So, the maximum term here <= 69. Indeed, a(1904738075) = a(380947615*10/2) = 69.

Below are all ending terms, along with the first, last, and number of terms in A010784 that lead to it:

End First Last Number

65 65 6890134752 1730

104 1 9876542130 7711661

130 130 9874521630 125045

164 41 9876543210 602389

208 208 208 1

260 260 260 1

328 328 328 1

416 416 9876245310 95141

520 321 9874621053 341720

832 832 832 1

The above all arise since the single step map x -> A137564(2*x) here has a closed orbit:

65 -> 130 -> 260 -> 520 -> 104 -> 208 -> 416 -> 832 -> 164 -> 328 -> 65.

(End)

Links

Table of n, a(n) for n=1..68.

Example

a(1) = 30 because the sequence is 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 653, 1306, 261, 52, 104, 208, 416, 832, 164, 328, 65, 130, 260, 520 and then repeats 104.
a(3) = 29 because the sequence is 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 614, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 653, 1306, 261, 52, 104, 208, 416, 832, 164, 328, 65, 130, 260, 520 and then repeats 104.
a(41) = 12 because the sequence is 41, 82, 164, 328, 65, 130, 260, 520, 104, 208, 416, 832 and then repeats 164.

Mathematica

a[n_] := -1 + Length@ NestWhileList[FromDigits[DeleteDuplicates[IntegerDigits[2*#]]] &, n, UnsameQ, All]; Array[a, 100] (* Amiram Eldar, Jan 01 2026 *)

Prog

(Python)
def f(n): # A137564
    seen, out, s = set(), "", str(n)
    for d in s:
        if d not in seen: out += d; seen.add(d)
    return int(out)
def a(n):
    seen = set()
    while n not in seen:
        seen.add(n)
        n = f(2*n)
    return len(seen)
print([a(n) for n in range(1, 69)]) # Michael S. Branicky, Jan 01 2026

Crossrefs

Cf. A137564, A392142.

Sequence in context: A056998 A057348 A057350 * A244948 A282130 A125564

Adjacent sequences: A392155 A392156 A392157 * A392159 A392160 A392161

Keyword

nonn,base

Author

Rodolfo Kurchan, Jan 01 2026

Status

approved