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%I #13 Feb 12 2022 20:30:42
%S 1,2,2,3,4,5,2,3,3,4,5,6,5,6,2,5,4,3,4,3,3,4,5,4,4,7,12,6,7,4,4,4,4,4,
%T 4,4,5,8,8,6,5,12,5,5,5,8,10,6,6,6,7,6,7,7,8,12,8,12,2,3,6,3,3,4,4,5,
%U 4,4,4,8,5,5,6,12,6,3,3,7,3,3,10,3,4,5,5,4,4,6,4,4,4,6,5,5,6,5,9,5,6,10,7,7,7
%N Least k such that the k-th arithmetic derivative of A351255(n) is zero.
%C a(n) is the number of iterations of the map x -> A003415(x) needed to reach zero, when starting from x = A351255(n).
%H Antti Karttunen, <a href="/A351257/b351257.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 a(n) = A099307(A351255(n)).
%F For all n, a(n) > A351256(n). [See A351258 for the differences].
%e From A351255(27) = 2625 it takes 12 iterations of map x -> A003415(x) to reach zero, as 2625 -> 2825 -> 1155 -> 886 -> 445 -> 94 -> 49 -> 14 -> 9 -> 6 -> 5 -> 1 -> 0, therefore a(27) = 12.
%o (PARI)
%o 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 A099307(n) = { my(s=1); while(n>1, n = A003415checked(n); s++); if(n,s,0); };
%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o for(n=0, 2^9, u=A276086(n); c = A099307(u); if(c>0,print1(c, ", ")));
%Y Cf. A003415, A099307, A276086, A351255, A351256, A351258, A351259, A351261.
%K nonn,look
%O 1,2
%A _Antti Karttunen_, Feb 11 2022
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