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a(1) = 1; for n > 1, a(n) is the least positive integer not occurring earlier that shares a digit but not a factor > 1 with a(n-1).
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%I #11 Mar 11 2021 20:38:07

%S 1,10,11,12,13,3,23,2,21,16,15,14,17,7,27,20,29,9,19,18,31,30,37,32,

%T 25,22,123,26,61,6,65,36,35,33,34,39,38,43,4,41,24,47,40,49,44,45,46,

%U 63,53,5,51,50,57,52,55,54,59,56,67,60,101,70,71,72,73,74,75,58,81,8,83,28,85,48

%N a(1) = 1; for n > 1, a(n) is the least positive integer not occurring earlier that shares a digit but not a factor > 1 with a(n-1).

%C After 100000 terms the lowest unused number is 99986. It is almost certain that this sequence is a permutation of the positive integers.

%H Scott R. Shannon, <a href="/A342367/a342367.png">Image of the first 1000 terms</a>. The green line is a(n) = n.

%t Block[{a = {1}, m = {1}, k}, Do[k = 2; While[Nand[FreeQ[a, k], GCD[k, a[[-1]]] == 1, IntersectingQ[m, IntegerDigits[k]]], k++]; AppendTo[a, k]; Set[m, IntegerDigits[k]], {i, 74}]; a] (* _Michael De Vlieger_, Mar 11 2021 *)

%o (Python)

%o from sympy import factorint

%o def aupton(terms):

%o alst, aset = [1], {1}

%o for n in range(2, terms+1):

%o an = 1

%o anm1_digs, anm1_factors = set(str(alst[-1])), set(factorint(alst[-1]))

%o while True:

%o while an in aset: an += 1

%o if set(str(an)) & anm1_digs != set():

%o if set(factorint(an)) & anm1_factors == set():

%o alst.append(an); aset.add(an); break

%o an += 1

%o return alst

%o print(aupton(74)) # _Michael S. Branicky_, Mar 09 2021

%Y Cf. A342356 (share factor and digit), A342366 (share factor but not digit), A239664 (no shared factor or digit), A184992, A309151, A249591.

%K nonn,base

%O 1,2

%A _Scott R. Shannon_, Mar 09 2021