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A351258 a(n) = A099307(A351255(n)) - A051903(A351255(n)).

%I A351258 #12 Feb 12 2022 17:47:12
%S A351258 1,1,1,2,2,3,1,2,2,2,3,3,2,2,1,4,3,2,2,2,2,2,3,2,2,4,9,4,5,2,2,2,2,2,
%T A351258 2,2,2,4,4,2,2,9,2,2,2,5,6,2,2,2,3,2,2,2,2,6,2,6,1,2,5,2,2,2,2,3,2,2,
%U A351258 2,5,2,2,2,8,5,2,2,6,2,2,9,2,2,3,3,2,2,2,2,2,2,4,2,2,2,2,6,2,3,7,3,3,3,4,2,2,3,2,8
%N A351258 a(n) = A099307(A351255(n)) - A051903(A351255(n)).
%C A351258 All terms are > 0 because from any k > 0, one certainly cannot reach 1 in less than A051903(k) iterations of the map x -> A003415(x).
%C A351258 One of the records occur at a(20457) = 38. The corresponding term of A351255 is A351255(20457) = A276086(A328116(20457)) = A276086(688352) = 442600020398400142264711707660915237 = 3 * 7^6 * 11^10 * 13^11 * 17^5 * 19. When starting iterating from this value with A003415, it first goes relatively smoothly in 11 steps to the first squarefree number encountered, 6201461846617177861789236821121654153, but after that, it still meanders for the additional 37 iterations (visiting mostly squarefree numbers, but also six numbers with max. exponent = 2, and one number with max. exponent = 3), before finally reaching zero.
%H A351258 Antti Karttunen, <a href="/A351258/b351258.txt">Table of n, a(n) for n = 1..105368</a> (computed for all 19-smooth terms of A351255, and also for A276086(9699690) = 23)
%F A351258 a(n) = A351257(n) - A351256(n) = A099307(A351255(n)) - A051903(A351255(n)).
%o A351258 (PARI)
%o A351258 A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
%o A351258 A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
%o A351258 A099307(n) = { my(s=1); while(n>1, n = A003415checked(n); s++); if(n,s,0); };
%o A351258 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A351258 for(n=0, 2^9, u=A276086(n); c = A099307(u); if(c>0,print1(c-A051903(u), ", ")));
%Y A351258 Cf. A003415, A051903, A099307, A351255, A351256, A351257.
%K A351258 nonn
%O A351258 1,4
%A A351258 _Antti Karttunen_, Feb 11 2022