A325446 The unitary version of Kalmár's function: number of ordered factorizations of n into powers of distinct primes.

1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 13, 1, 1, 3, 3, 3, 3, 1, 3, 3, 3, 1, 13, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 13, 1, 3, 3, 1, 3, 13, 1, 3, 3, 13, 1, 3, 1, 3, 3, 3, 3, 13, 1, 3, 1, 3, 1, 13, 3

Offset: 1

Author

Amiram Eldar, Sep 06 2019

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221, A000670, A005117, A074206.

Programs

Mathematica

f[0] = 1; f[n_] := f[n] = Sum[Binomial[n, k] * f[n - k], {k, 1, n}]; a[n_] := f[PrimeNu[n]]; Array[a, 100]

Formula

a(1) = 1 and a(n) = Sum_{d||n} a(d), where d||n are the unitary divisors of n (d such that gcd(d, n/d) = 1).

a(n) = A000670(A001221(n)).

a(n) = A074206(n) for squarefree n (A005117).