A380160 a(n) is the value of the Euler totient function when applied to the powerful part of n.

1, 1, 1, 2, 1, 1, 1, 4, 6, 1, 1, 2, 1, 1, 1, 8, 1, 6, 1, 2, 1, 1, 1, 4, 20, 1, 18, 2, 1, 1, 1, 16, 1, 1, 1, 12, 1, 1, 1, 4, 1, 1, 1, 2, 6, 1, 1, 8, 42, 20, 1, 2, 1, 18, 1, 4, 1, 1, 1, 2, 1, 1, 6, 32, 1, 1, 1, 2, 1, 1, 1, 24, 1, 1, 20, 2, 1, 1, 1, 8, 54, 1, 1, 2

Offset: 1

Author

Amiram Eldar, Jan 13 2025

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A000010, A001694, A005117, A057521, A295294, A295295, A323333, A349379, A357669, A380161.

Programs

Mathematica

f[p_, e_] := If[e == 1, 1, (p-1)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, (f[i, 1]-1)*f[i, 1]^(f[i, 2]-1)));}

Formula

a(n) = A000010(A057521(n)).

a(n) >= 1, with equality if and only if n is squarefree (A005117).

a(n) <= A000010(n), with equality if and only if n is powerful (A001694).

Multiplicative with a(p) = 1, and a(p^e) = (p-1)*p^(e-1) if e >= 2.

Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(s-1) + 1/p^(2*s-2) - 2/p^(2*s-1)).

Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = Product_{p prime} (1 + 2/p^(3/2) - 1/p^2 - 2/p^(5/2)) = 1.96428740396979919886... .