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a(1) = 1, and for n > 1, a(n) is the largest prime factor (A006530) of all terms encountered when iterating the map x -> A000593(x), when starting from x = n, and including the n itself. If 1 is never reached when starting from n, then a(n) = -1.
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%I #8 Feb 10 2022 20:05:57

%S 1,2,3,2,5,3,7,2,13,5,11,3,13,7,5,2,17,13,19,5,7,11,23,3,31,13,5,7,29,

%T 5,31,2,11,17,7,13,37,19,13,5,41,7,43,11,13,23,47,3,19,31,17,13,53,5,

%U 13,7,19,29,59,5,61,31,13,2,13,11,67,17,23,7,71,13,73,37,31,19,11,13,79,5,19,41,83,7,17,43,29

%N a(1) = 1, and for n > 1, a(n) is the largest prime factor (A006530) of all terms encountered when iterating the map x -> A000593(x), when starting from x = n, and including the n itself. If 1 is never reached when starting from n, then a(n) = -1.

%H Antti Karttunen, <a href="/A347241/b347241.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = max(A006530(n), A347240(n)).

%e For n = 17, the iteration proceeds as follows 17 -> 18 (= 2*3*3), 18 -> 13 (13 is a prime), 13 -> 14 (= 2*7), 14 -> 8 (= 2*2*2), 8 -> 1. The largest prime factor present (when including the starting term also) is 17, thus a(17) = 17.

%o (PARI)

%o A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);

%o A000265(n) = (n >> valuation(n, 2));

%o A000593(n) = sigma(A000265(n));

%o A347241(n) = { my(m=1); while(n>1, m = max(m, A006530(n)); n = A000593(n)); (m); };

%Y Cf. A000593, A006530, A161942, A336361, A347240, A347242.

%K nonn

%O 1,2

%A _Antti Karttunen_, Aug 28 2021