A308427 a(n) is the number of central factorizations needed to reach the prime factorization of n.

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

Offset

1,8

Comments

The central factorization of a positive integer m is m*(n/m), where m is the greatest divisor of n that is <= sqrt(n).

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Example

32 = 4*8 = (2*2)*(2*4) = (2*2)*(2*(2*2)), so that a(32) = 3.

Mathematica

f[n_] := Last[Select[Divisors[n], # <= Sqrt[n] &]];
a[1] = 0; a[2] = 0; a[n_] := If[f[n] == 1, 0, 1 + Max[a[f[n]], a[n/f[n]]]];
Table[a[n], {n, 1, 60}]

Crossrefs

Cf. A000040.

Sequence in context: A257857 A339871 A276806 * A252736 A351416 A253559

Adjacent sequences: A308424 A308425 A308426 * A308428 A308429 A308430

Keyword

nonn,easy

Author

Clark Kimberling, Jun 03 2019

Status

approved