A380164 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is a square.

1, 1, 1, 2, 1, 1, 1, 1, 6, 1, 1, 2, 1, 1, 1, 8, 1, 6, 1, 2, 1, 1, 1, 1, 20, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 8, 42, 20, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 6, 32, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 20, 2, 1, 1, 1, 8, 54, 1, 1, 2, 1, 1, 1, 1, 1, 6, 1, 2, 1, 1, 1, 1, 1, 42, 6, 40

Offset

1,4

Links

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

Formula

a(n) = A000010(A350388(n)).

a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).

a(n) <= A000010(n), with equality if and only if n is either a square (A000290) or twice an odd square (A077591 \ {1}).

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

Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s-1) - 1/p^(2*s) - 1/p^(3*s-2) + 1/p^(4*s-1)).

Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^2 - 1/p^(5/2) - 1/p^3 + 1/p^5) = 1.16404670858123447768... .

Mathematica

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

Crossrefs

Cf. A000010, A000290, A002117, A077591, A268335, A350388, A351568, A365401, A380165.

Sequence in context: A124341 A275062 A247005 * A174215 A364457 A305567

Adjacent sequences: A380161 A380162 A380163 * A380165 A380166 A380167

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 14 2025

Status

approved