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).

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, 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, 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

Offset: 1

Author

Antti Karttunen, Oct 13 2019

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A003415, A051903, A328311.

Cf. A008578 (positions of 1's), A099309 (of 0's).

Cf. also A328248.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328311(n) = if(n<=1,0,1+(A051903(A003415(n)) - A051903(n)));
A328312(n) = { my(m=1); while(n>1, m *= (1+A328311(n)); n = A003415checked(n)); (n*m); };