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
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
-
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]
-
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)));}
Formula
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... .