A380165 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is an exponentially odd number.

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

Offset

1,3

Links

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

Formula

a(n) = A000010(A350389(n)).

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

a(n) <= A000010(n), with equality if and only if n is an exponentially odd number (A268335).

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

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^5) = 0.50115112192510092436... .

Mathematica

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

Crossrefs

Cf. A000010, A000290, A013662, A077591, A268335, A350389, A351569, A365402, A374456, A380164.

Sequence in context: A065423 A239242 A340621 * A008733 A359907 A244515

Adjacent sequences: A380162 A380163 A380164 * A380166 A380167 A380168

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 14 2025

Status

approved