A019269 Let Dedekind's psi(m) = product of (p+1)p^(e-1) for primes p, where p^e is a factor of m. Iterating psi(m) eventually results in a number of form 2^a*3^b. a(n) is the number of steps to reach such a number.

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 1, 1, 0, 2, 2, 0, 1, 2, 1, 1, 0, 1, 1, 1, 0, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 0, 2, 2, 1, 2, 1, 0, 1, 1, 2, 2, 2, 1, 2, 1, 1, 0, 2, 1, 2, 1, 1, 1, 1, 0, 4, 3, 2, 2, 1, 2, 2, 1, 0, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 0, 3, 2, 1

Offset: 1

Author

Jud McCranie

Keywords

• nonn

Comments

a(A019268(n)) = n and a(m) <> n for m < A019268(n). [Reinhard Zumkeller, Apr 12 2012]

References

• Peter Giblin, "Primes and Programming - an Introduction to Number Theory with Computation", page 118.
• R. K. Guy, "Unsolved Problems in Number Theory", section B41.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001615, A065333.

Programs

Haskell

a019269 n = snd $ until ((== 1) . a065333 . fst)
                        (\(x, i) -> (a001615 x, i+1)) (n, 0)
-- Reinhard Zumkeller, Apr 12 2012

Mathematica

psi[n_] := Module[{pp, ee}, {pp, ee} = Transpose[FactorInteger[n]]; If[Max[pp] == 3, n, Times @@ (pp+1)*Times @@ (pp^(ee-1))]];
a[n_] := Length[NestWhileList[psi, n, FactorInteger[#][[-1, 1]] > 3&]] - 1;
a /@ Range[99] (* Jean-François Alcover, Jan 18 2020 *)