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A355061 Lexicographically earliest infinite sequence of positive numbers such that, for n>2, a(n) has a common factor with a(n-1), no common factor with a(n-2), and the product a(n)*a(n-1) is distinct from all previous products, a(i)*a(i-1), i=2..n-1.

%I A355061 #11 Jun 21 2022 10:28:52
%S A355061 1,2,6,15,35,14,6,33,55,10,6,21,35,10,12,21,77,22,6,39,65,10,14,21,15,
%T A355061 10,22,33,15,20,14,63,15,40,14,77,33,12,14,91,39,12,20,35,63,6,26,65,
%U A355061 15,12,22,55,15,18,28,35,45,12,26,91,21,30,22,143,39,15,50,22,99,15,70,22,187,51,6
%N A355061 Lexicographically earliest infinite sequence of positive numbers such that, for n>2, a(n) has a common factor with a(n-1), no common factor with a(n-2), and the product a(n)*a(n-1) is distinct from all previous products, a(i)*a(i-1), i=2..n-1.
%C A355061 Like the Enots Wolley sequence, A336957, no term a(n) can be a prime or a prime power as this would make it impossible to find a(n+1). As 6 is the smallest number to include two different primes, and hence the smallest number beyond the first two terms that can appear, it occurs frequently in the sequence, 1887 times in the first 250000 terms. See A355139 for the indices of these terms.
%C A355061 Unlike A336957 multiple odd successive terms occur, the longest such run in the first 250000 terms being fourteen starting at a(111799) = 20257.
%C A355061 See A355138 for the products of consecutive terms.
%H A355061 Scott R. Shannon, <a href="/A355061/a355061.png">Image of the first 250000 terms</a>. The green line is y = n.
%e A355061 a(5) = 35 as this is the smallest number to share a factor with a(4) = 15, not share a factor with a(3) = 6, and contains a prime factor not in a(4) = 15 and hence allows a(6) to exist.
%e A355061 a(7) = 6 as this is the smallest number to share a factor with a(6) = 14, not share a factor with a(5) = 35, and contains a prime factor not in a(6) = 14 and hence allows a(8) to exist. This is the first term to differ from A336957.
%o A355061 (Python)
%o A355061 from sympy import primefactors
%o A355061 from itertools import count, islice
%o A355061 def agen(): # generator of terms
%o A355061     an1, an, f1, f, pset = 2, 6, {2}, {2, 3}, {2, 12}
%o A355061     yield from [1, 2, 6]
%o A355061     for n in count(4):
%o A355061         an2, an1, an, f2, f1 = an1, an, 6, f1, f
%o A355061         f = set(primefactors(an))
%o A355061         while an*an1 in pset or f1&f == set() or f2&f != set() or f <= f1:
%o A355061             an += 1; f = set(primefactors(an))
%o A355061         pset.add(an*an1); yield an
%o A355061 print(list(islice(agen(), 75))) # _Michael S. Branicky_, Jun 20 2022
%Y A355061 Cf. A355138, A355139, A336957, A098550, A064413.
%K A355061 nonn
%O A355061 1,2
%A A355061 _Scott R. Shannon_, Jun 16 2022