A378929 Number of steps it takes for the chain of f(n) = n * d(n) / gcd(n, d(n))^2 to reach a cycle from a starting number n.

0, 1, 1, 3, 1, 0, 1, 2, 2, 0, 1, 2, 1, 0, 3, 4, 1, 2, 1, 4, 3, 0, 1, 2, 1, 0, 4, 4, 1, 3, 1, 6, 3, 0, 1, 4, 1, 0, 3, 2, 1, 3, 1, 4, 4, 0, 1, 5, 1, 0, 3, 4, 1, 4, 1, 2, 3, 0, 1, 2, 1, 0, 4, 8, 1, 3, 1, 4, 3, 0, 1, 1, 1, 0, 0, 4, 1, 3, 1, 3, 1, 0, 1, 2, 1, 0, 3, 2

Offset: 1

Author

Viliam Furík, Dec 11 2024

Keywords

• nonn

Comments

It is conjectured that every number n eventually ends up in a cycle.

Links

• Viliam Furík, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A378930.

Programs

Mathematica

f[n_] := Module[{d = DivisorSigma[0, n]}, n*d/GCD[n, d]^2]; a[n_] := Module[{s = NestWhileList[f, n, UnsameQ, All]}, FirstPosition[s, s[[-1]]][[1]] - 1]; Array[a, 100] (* Amiram Eldar, Dec 16 2024 *)