A328248 a(n) = 1 if n is a squarefree number (A005117), otherwise a(n) = 1 + number of iterations of arithmetic derivative (A003415) needed to reach a squarefree number, or 0 if no such number is ever reached.

1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 2, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 2, 3, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 2, 3, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0

Offset

1,9

Links

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

Formula

a(4*n) = a(27*n) = 0 and in general, a(m * p^p) = 0, for any m >= 1 and any prime p.

Example

For n = 9, it itself is not a squarefree number, while its arithmetic derivative A003415(9) = 6 is, so it took just one iteration to find a squarefree number, thus a(9) = 1+1 = 2.
For n = 50, which is not squarefree, and its first derivative A003415(50) = 45 also is not squarefree, but taking derivative yet again, gives A003415(45) = 39 = 3*13, which is squarefree, thus a(50) = 2+1 = 3.

Prog

(PARI)
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));
A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };

Crossrefs

Cf. A003415, A008966, A256750.

Cf. A328251, A005117, A328252, A328253 (indices of terms k=0, 1, 2, 3).

Sequence in context: A336387 A358217 A286133 * A360158 A329885 A085737

Adjacent sequences: A328245 A328246 A328247 * A328249 A328250 A328251

Keyword

nonn

Author

Antti Karttunen, Oct 11 2019

Status

approved