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

1,4

Links

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

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... .

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]

Prog

(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))); }

Crossrefs

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

Sequence in context: A095231 A303697 A362827 * A342413 A202019 A295685

Adjacent sequences: A380157 A380158 A380159 * A380161 A380162 A380163

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 13 2025

Status

approved