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A350444 a(n) is the smallest number that has not appeared yet in the sequence and has only one digit in common with a(n-1).

%I A350444 #37 Jan 07 2022 15:49:58
%S A350444 1,10,12,2,20,21,13,3,23,24,4,14,15,5,25,26,6,16,17,7,27,28,8,18,19,9,
%T A350444 29,32,30,31,34,35,36,37,38,39,43,40,41,42,45,46,47,48,49,54,50,51,52,
%U A350444 53,56,57,58,59,65,60,61,62,63,64,67,68,69,76,70,71,72,73,74,75,78,79,87,80,81,82,83,84,85,86,89,90
%N A350444 a(n) is the smallest number that has not appeared yet in the sequence and has only one digit in common with a(n-1).
%C A350444 Terms computed by _Claudio Meller_.
%C A350444 It appears that this sequence contains all numbers except those in A115853.
%H A350444 Michael De Vlieger, <a href="/A350444/a350444.png">Log-log scatterplot of a(n)</a> for n = 1..10000 labeling the first 40 terms.
%e A350444 a(2) = 10 because it is the smallest number that has exactly one digit in common with a(1) = 1; similarly, a(3) = 12 because it has one digit in common with a(2) = 10 and a(4) = 2 because it is the smallest number that is not already in the sequence that has exactly one digit in common with a(3) = 12.
%t A350444 c[_] = False; j = c[1] = 1; {j}~Join~Reap[Do[d = Union@ IntegerDigits[j]; If[j == u, While[c[u] > 0, u++]]; k = u; While[Nand[c[k] == False, Count[IntegerDigits[k], _?(MemberQ[d, #] &)] == 1], k++]; Sow[k]; c[k] = True; j = k, 81]][[-1, -1]] (* _Michael De Vlieger_, Dec 31 2021 *)
%o A350444 (Python)
%o A350444 from itertools import islice
%o A350444 def c(s, t): return sum(t.count(si) for si in s)
%o A350444 def agen(): # generator of terms
%o A350444     an, target, seen, mink = 1, "1", {1}, 2
%o A350444     while True:
%o A350444         yield an
%o A350444         k = mink
%o A350444         while k in seen or c(str(k), target) != 1: k += 1
%o A350444         an, target = k, str(k)
%o A350444         seen.add(an)
%o A350444         while mink in seen: mink += 1
%o A350444 print(list(islice(agen(), 82))) # _Michael S. Branicky_, Dec 31 2021
%Y A350444 Cf. A115853, A350445.
%K A350444 nonn,base
%O A350444 1,2
%A A350444 _Rodolfo Kurchan_, Dec 31 2021