A390755 The Euler totient of the smallest powerful number divisible by n.

1, 2, 6, 2, 20, 12, 42, 4, 6, 40, 110, 12, 156, 84, 120, 8, 272, 12, 342, 40, 252, 220, 506, 24, 20, 312, 18, 84, 812, 240, 930, 16, 660, 544, 840, 12, 1332, 684, 936, 80, 1640, 504, 1806, 220, 120, 1012, 2162, 48, 42, 40, 1632, 312, 2756, 36, 2200, 168, 2052, 1624

Offset

1,2

Comments

Differs from A327171 at n = 8, 24, 27, 32, 40, 54, 56, ... .

Links

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

Formula

a(n) = A000010(A197863(n)).

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

Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + 1/p^(s-2) - 2/p^(s-1) - 1/p^(2*s-3) + 2/p^(2*s-2) - 1/p^(2*s-1)).

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * Product_{p prime} (1 - 3/p^2 + 1/p^3 + 3/p^4 - 3/p^5 + 1/p^6) = 0.44658498822682177228... .

Sum_{n>=1} 1/a(n) = zeta(2)^2 * Product_{p prime} (1 + 3/p^3 + 1/p^4 - 1/p^5) = 4.43107782971081507625... .

Mathematica

f[p_, e_] := (p-1) * p^If[e == 1, 1, 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~, (f[i, 1] - 1) * f[i, 1]^if(f[i, 2] == 1, 1, f[i, 2] - 1)); }

Crossrefs

Cf. A000010, A013661, A197863, A327171, A390756.

Sequence in context: A174857 A248568 A257252 * A327171 A008556 A254638

Adjacent sequences: A390752 A390753 A390754 * A390756 A390757 A390758

Keyword

nonn,mult,easy

Author

Amiram Eldar, Nov 17 2025

Status

approved