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A342755 a(1) = 2; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n) shares no digit with a(n-1) and a(n-1)*a(n) shares no digit with either a(n-1) or a(n).

%I A342755 #18 Apr 11 2023 02:17:50
%S A342755 2,3,4,5,6,7,8,9,42,15,22,14,55,12,37,16,25,36,29,47,23,46,13,44,18,
%T A342755 32,17,38,19,33,26,35,174,53,76,59,34,27,43,67,49,62,87,106,493,57,24,
%U A342755 75,48,65,122,39,54,72,88,45,66,73,56,77,52,79,84,63,78,123,69,58,64,92,74,68,114,85,314
%N A342755 a(1) = 2; for n > 1, a(n) is the least positive integer not occurring earlier such that a(n) shares no digit with a(n-1) and a(n-1)*a(n) shares no digit with either a(n-1) or a(n).
%C A342755 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. Currently the last known term is a(173) = 922989, the next being at least 5*10^10 if it exists. It is unknown if the sequence is infinite.
%C A342755 a(174) = 60060666070067077 and a(175) has 52 digits (see b-file). If a(176) exists, it is > 10^71. - _Michael S. Branicky_, Apr 10 2023
%H A342755 Michael S. Branicky, <a href="/A342755/b342755.txt">Table of n, a(n) for n = 1..175</a>
%e A342755 a(2) = 3 as 3 shares no digit with a(1) = 2 and a(1)*3 = 2*3 = 6 shares no digit with a(1) = 2 or 3.
%e A342755 a(9) = 42 as 42 shares no digit with a(8) = 9 and a(8)*42 = 9*42 = 378 shares no digit with a(8) = 9 or 42.
%e A342755 a(10) = 15 as 15 shares no digit with a(9) = 42 and a(9)*15 = 42*15 = 630 shares no digit with a(9) = 42 or 15. This is the first term that differs from A342442.
%e A342755 a(173) = 922989 as 922989 shares no digit with a(172) = 7154 and a(172)*922989 = 7154*922989 = 6603063306 shares no digit with a(172) = 7154 or 922989. This is currently the last known term.
%o A342755 (Python)
%o A342755 def aupton(terms):
%o A342755   alst, aset = [2], {2}
%o A342755   while len(alst) < terms:
%o A342755     an, anm1_digs = 2, set(str(alst[-1]))
%o A342755     while True:
%o A342755       while an in aset: an += 1
%o A342755       an_digs = set(str(an))
%o A342755       if (an_digs & anm1_digs) == set():
%o A342755         prod_digs = set(str(an*alst[-1]))
%o A342755         if (anm1_digs | an_digs) & prod_digs  == set():
%o A342755           alst.append(an); aset.add(an); break
%o A342755       an += 1
%o A342755   return alst
%o A342755 print(aupton(173)) # _Michael S. Branicky_, Mar 21 2021
%Y A342755 Cf. A342442, A342441, A276633, A010784, A043537, A043096, A338466, A336285.
%K A342755 nonn,base
%O A342755 1,1
%A A342755 _Scott R. Shannon_, Mar 20 2021