A353888 a(n) is the least positive integer not occurring earlier in the sequence that contains at least one digit not in a(n-1); a(1)=1.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 11, 21, 23, 24, 25, 26, 27, 28, 29, 30, 22, 31, 32, 34, 35, 36, 37, 38, 39, 40, 33, 41, 42, 43, 45, 46, 47, 48, 49, 50, 44, 51, 52, 53, 54, 56, 57, 58, 59, 60, 55, 61, 62, 63, 64, 65, 67, 68

Offset: 1

Author

Sergio Pimentel, May 09 2022

Keywords

• nonn
• base
• fini

Comments

The sequence is finite. 1023456789 should be the last number in the sequence, although many smaller numbers should fail to appear. How many terms are in the complete sequence?

The last term is a(1023445778) = 1023456789, the least missing number is 1000000010. - Rémy Sigrist, Jun 03 2022

At that point, the least missing numbers containing the digits 2..9 are 1020001000, 1023001300, 1023401000, 1023450200, 1023456024, 1023456710, 1023456781, 1023456789, resp. - Michael S. Branicky, Aug 26 2022

Examples

			a(11)=12 since a(10)=10 and 12 is the smallest number not occurring earlier in the sequence that contains a digit (2) that is not in 10.
		

Links

• Sergio Pimentel, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, C++ program

Crossrefs

Cf. A184992, A162501, A130571, A107353.

Programs

PARI

isok(k, prev) = {my(d=digits(k)); for (i=1, #d, if (!vecsearch(prev, d[i]), return(1));); return(0);}
find(va, n) = {my(k=1, prev=Set(digits(va[n-1]))); while (vecsearch(Set(va), k) || !isok(k, prev), k++); k;}
lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = find(va, n);); va;} \\ Michel Marcus, May 11 2022
(C++) See Links section.

Python

from itertools import count, islice
def agen():  # generator of terms
    an, aset, mu, mink = 0, set(), [10, 1, 2, 3, 4, 5, 6, 7, 8, 9], 1
    while set(str(an)) != set("0123456789"):
        notin = set("0123456789") - set(str(an))
        an = min(mu[i] for i in range(10) if str(i) in notin)
        yield an; aset.add(an)
        for i in range(10):  # update min unused containing digit i
            while mu[i] in aset or str(i) not in str(mu[i]): mu[i] += 1
        for k in range(mink, min(mu)): aset.discard(k)
        mink = min(mu)
print(list(islice(agen(), 67))) # Michael S. Branicky, Aug 26 2022