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A328312 a(n) is the product of (1+A328311(x)) applied over all values x obtained when arithmetic derivative (A003415) is iterated starting from x=n, until 1 is encountered, or 0 if no 1 is ever encountered (in which case such product would be infinite).

%I A328312 #11 Oct 13 2019 18:09:23
%S A328312 1,1,1,0,1,2,1,0,2,2,1,0,1,6,0,0,1,4,1,0,4,2,1,0,2,0,0,0,1,2,1,0,12,2,
%T A328312 0,0,1,8,0,0,1,2,1,0,0,6,1,0,6,0,0,0,1,0,0,0,4,2,1,0,1,24,0,0,12,2,1,
%U A328312 0,0,2,1,0,1,0,0,0,12,2,1,0,0,2,1,0,4,0,0,0,1,0,0,0,4,18,0,0,1,12,0,0,1,0,1,0,2
%N A328312 a(n) is the product of (1+A328311(x)) applied over all values x obtained when arithmetic derivative (A003415) is iterated starting from x=n, until 1 is encountered, or 0 if no 1 is ever encountered (in which case such product would be infinite).
%H A328312 Antti Karttunen, <a href="/A328312/b328312.txt">Table of n, a(n) for n = 1..65537</a>
%o A328312 (PARI)
%o A328312 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A328312 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 A328312 A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
%o A328312 A328311(n) = if(n<=1,0,1+(A051903(A003415(n)) - A051903(n)));
%o A328312 A328312(n) = { my(m=1); while(n>1, m *= (1+A328311(n)); n = A003415checked(n)); (n*m); };
%Y A328312 Cf. A003415, A051903, A328311.
%Y A328312 Cf. A008578 (positions of 1's), A099309 (of 0's).
%Y A328312 Cf. also A328248.
%K A328312 sign
%O A328312 1,6
%A A328312 _Antti Karttunen_, Oct 13 2019