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%I #21 Mar 24 2021 23:05:37
%S 2,3,4,5,6,7,8,9,42,14,17,18,15,16,13,19,32,22,23,26,29,12,25,24,34,
%T 27,33,36,39,43,37,38,28,47,44,45,46,63,66,65,48,49,62,55,54,35,174,
%U 53,76,57,56,59,52,58,64,92,74,68,78,72,77,67,73,83,69,79,84,75,88,113,183,138,149,148
%N a(1) = 2; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n-1)*a(n) shares no digit with either a(n-1) or a(n).
%C No term can end in 0 or 1 as that would result in the last digit of a(n-1)*a(n) being the same as a(n)'s last digit. The majority of terms appear to grow linearly with n but occasional large spikes in the values also occur, e.g. a(47888) = 425956849. See the examples. It is unknown if the sequence is infinite.
%e a(2) = 3 as a(1)*3 = 2*3 = 6 which shares no digit with a(1) = 2 or 3.
%e a(9) = 42 as a(8)*42 = 9*42 = 378 which shares no digit with a(8) = 9 or 42.
%e a(10) = 14 as a(9)*14 = 42*14 = 588 which shares no digit with a(9) = 42 or 14.
%e a(47888) = 425956849 as a(47887)*425956849 = 258649*425956849 = 110173313037001 which shares no digit with a(47887) = 258649 or 425956849.
%o (Python)
%o def aupton(terms):
%o alst, aset = [2], {2}
%o while len(alst) < terms:
%o an, anm1_digs = 2, set(str(alst[-1]))
%o while True:
%o while an in aset: an += 1
%o if (set(str(an)) | anm1_digs) & set(str(an*alst[-1])) == set():
%o alst.append(an); aset.add(an); break
%o an += 1
%o return alst
%o print(aupton(74)) # _Michael S. Branicky_, Mar 20 2021
%Y Cf. A342441 (addition), A342755, A276633, A010784, A043537, A043096, A338466, A336285.
%K nonn,base
%O 1,1
%A _Scott R. Shannon_, Mar 12 2021
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