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A342356 a(1) = 1, a(2) = 10; for n > 2, a(n) is the least positive integer not occurring earlier that shares both a factor and a digit with a(n-1).

%I A342356 #21 Mar 11 2021 20:36:42
%S A342356 1,10,12,2,20,22,24,4,14,16,6,26,28,8,18,15,5,25,35,30,3,33,36,32,34,
%T A342356 38,48,40,42,21,27,57,45,50,52,54,44,46,56,58,68,60,62,64,66,63,39,9,
%U A342356 69,90,70,7,77,147,49,84,74,37,333,93,31,124,72,75,51,17,102,80,78,76,86,82,88,98,91
%N A342356 a(1) = 1, a(2) = 10; for n > 2, a(n) is the least positive integer not occurring earlier that shares both a factor and a digit with a(n-1).
%C A342356 After 100000 terms the lowest unused number is 18181. The sequence is likely a permutation of the positive integers.
%H A342356 Scott R. Shannon, <a href="/A342356/a342356.png">Image of the first 100000 terms</a>. The green line is a(n) = n.
%t A342356 Block[{a = {1, 10}, m = {1, 0}, 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, 73}]; a] (* _Michael De Vlieger_, Mar 11 2021 *)
%o A342356 (Python)
%o A342356 from sympy import factorint
%o A342356 def aupton(terms):
%o A342356   alst, aset = [1, 10], {1, 10}
%o A342356   for n in range(3, terms+1):
%o A342356     an = 1
%o A342356     anm1_digs, anm1_factors = set(str(alst[-1])), set(factorint(alst[-1]))
%o A342356     while True:
%o A342356       while an in aset: an += 1
%o A342356       if set(str(an)) & anm1_digs != set():
%o A342356         if set(factorint(an)) & anm1_factors != set():
%o A342356           alst.append(an); aset.add(an); break
%o A342356       an += 1
%o A342356   return alst
%o A342356 print(aupton(75)) # _Michael S. Branicky_, Mar 09 2021
%Y A342356 Cf. A342366 (share factor but not digit), A239664 (no shared factor or digit), A342367 (share digit but not factor), A184992, A309151, A249591.
%K A342356 nonn,base
%O A342356 1,2
%A A342356 _Scott R. Shannon_, Mar 08 2021