A380163 a(n) is the value of the Euler totient function when applied to the squarefree part of n.

1, 1, 2, 1, 4, 2, 6, 1, 1, 4, 10, 2, 12, 6, 8, 1, 16, 1, 18, 4, 12, 10, 22, 2, 1, 12, 2, 6, 28, 8, 30, 1, 20, 16, 24, 1, 36, 18, 24, 4, 40, 12, 42, 10, 4, 22, 46, 2, 1, 1, 32, 12, 52, 2, 40, 6, 36, 28, 58, 8, 60, 30, 6, 1, 48, 20, 66, 16, 44, 24, 70, 1, 72, 36

Offset

1,3

Links

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

Formula

a(n) = A000010(A007913(n)).

a(n) >= 1, with equality if and only if n is in A028982.

a(n) <= A000010(n), with equality if and only if n is squarefree (A005117).

Multiplicative with a(p^e) = p-1 if e is odd, and 1 otherwise.

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^2 + 1/p^3) = 0.46350438981962928756...

Mathematica

f[p_, e_] := If[OddQ[e], p-1, 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] % 2, f[i, 1]-1, 1)); }

Crossrefs

Cf. A000010, A005117, A007913, A013662, A028982, A055076, A367991, A380162.

Sequence in context: A342415 A098371 A380161 * A300234 A070777 A173614

Adjacent sequences: A380160 A380161 A380162 * A380164 A380165 A380166

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 14 2025

Status

approved