A348001 Number of distinct values obtained when the unitary totient function (A047994) is applied to the unitary divisors of n.

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 4, 4, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 4, 2, 2, 8, 2, 2, 4, 2, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 4, 4, 4, 4, 2, 4, 2, 2, 2, 7, 4, 2, 4

Offset

1,3

Links

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

Formula

a(2^e) = 2 for e > 1.

a(p^e) = 2 for an odd prime p and e > 0.

a(n) >= omega(n), with equality if and only if n is in A278568.

Example

n = 6 has four unitary divisors: 1, 2, 3 and 6. Applying A047994 to these gives 1, 1, 2 and 2, with just 2 distinct values, thus a(6) = 2.
n = 12 has four unitary divisors: 1, 3, 4 and 12. Applying A047994 to these gives 4 distinct values, 1, 2, 3 and 6, thus a(12) = 4.

Mathematica

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Length @ Union[uphi /@ Select[Divisors[n], CoprimeQ[#, n/#] &]]; Array[a, 100]

Crossrefs

The unitary version of A319696.

Cf. A047994, A077610, A278568.

Sequence in context: A343336 A103817 A073808 * A361923 A032572 A237975

Adjacent sequences: A347998 A347999 A348000 * A348002 A348003 A348004

Keyword

nonn

Author

Amiram Eldar, Sep 23 2021

Status

approved