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A365521 a(1) = 1; for n > 1, a(n) is the prime factor of n that has not appeared for the longest time in {a(1),...,a(n-2),a(n-1)}.

%I A365521 #41 Jan 20 2024 09:49:05
%S A365521 1,2,3,2,5,3,7,2,3,5,11,2,13,7,3,2,17,3,19,5,7,11,23,2,5,13,3,7,29,2,
%T A365521 31,2,11,17,5,3,37,19,13,2,41,7,43,11,5,23,47,3,7,2,17,13,53,3,11,7,
%U A365521 19,29,59,5,61,31,3,2,13,11,67,17,23,7,71,3,73,37,5,19
%N A365521 a(1) = 1; for n > 1, a(n) is the prime factor of n that has not appeared for the longest time in {a(1),...,a(n-2),a(n-1)}.
%H A365521 Jianglin Luo, <a href="/A365521/b365521.txt">Table of n, a(n) for n = 1..10080</a>
%H A365521 David A. Corneth, <a href="/A365521/a365521.gp.txt">PARI program</a>
%H A365521 Jianglin Luo, <a href="https://2293.github.io/mathdemos/100001historical%20prime%20factor.html">Sagecell</a>
%e A365521 a(6)=3 because 6 = 2*3 and 2=a(4) has appeared more recently than 3=a(3).
%e A365521 a(12)=2 because 12 = 2^2*3 and 3=a(9) has appeared more recently than 2=a(8).
%e A365521 a(30)=2 because 30 = 2*3*5 and 3=a(27) and 5=a(25) have appeared more recently than 2=a(24).
%o A365521 (SageMath)
%o A365521 def hpf_seq(top):
%o A365521     H=[0,1,2,3]
%o A365521     for n in range(4,top):
%o A365521         prime_factors=[part[0] for part in list(factor(n))]
%o A365521         cursor=-1
%o A365521         while len(prime_factors)>1:
%o A365521             if H[cursor] in prime_factors:
%o A365521                 prime_factors.remove(H[cursor])
%o A365521             cursor-=1
%o A365521         hpf=prime_factors[0]
%o A365521         H.append(hpf)
%o A365521     return H
%o A365521 (PARI) See PARI link \\ _David A. Corneth_, Sep 08 2023
%Y A365521 Cf. A088387, A006530, A034699, A088388.
%K A365521 nonn,easy
%O A365521 1,2
%A A365521 _Jianglin Luo_, Sep 08 2023