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

Author

Amiram Eldar, Jan 14 2025

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

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

Programs

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]

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));}

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... .