A380161 a(n) is the value of the Euler totient function when applied to the powerfree 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, 1, 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, 1, 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(A055231(n)).

a(n) >= 1, with equality if and only if n is in A335851.

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

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

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * Product_{p prime} (1 - 2/p^5 + 3/p^4 + 1/p^3 - 3/p^2) = 0.46288631864329056778... .

Mathematica

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

Crossrefs

Cf. A000010, A005117, A013661, A055231, A056671, A092261, A335851, A380160.

Sequence in context: A147763 A342415 A098371 * A380163 A300234 A070777

Adjacent sequences: A380158 A380159 A380160 * A380162 A380163 A380164

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 13 2025

Status

approved