A333569 a(n) = Sum_{d|n} (-1)^(bigomega(d) - omega(d)) * phi(n/d).
1, 2, 3, 2, 5, 6, 7, 6, 7, 10, 11, 6, 13, 14, 15, 10, 17, 14, 19, 10, 21, 22, 23, 18, 23, 26, 23, 14, 29, 30, 31, 22, 33, 34, 35, 14, 37, 38, 39, 30, 41, 42, 43, 22, 35, 46, 47, 30, 47, 46, 51, 26, 53, 46, 55, 42, 57, 58, 59, 30, 61, 62, 49, 42, 65, 66, 67, 34, 69, 70, 71, 42, 73, 74, 69
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
-
Mathematica
Table[Sum[(-1)^(PrimeOmega[d] - PrimeNu[d]) EulerPhi[n/d], {d, Divisors[n]}], {n, 1, 75}] Table[Sum[(-1)^(PrimeOmega[GCD[n, k]] - PrimeNu[GCD[n, k]]), {k, 1, n}], {n, 1, 75}] f[p_, e_] := If[e > 1, (p^e*(p^2+p-2) - 2*(-1)^e*p)/(p*(p + 1)), p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100] (* Amiram Eldar, Nov 12 2022 *) -
PARI
a(n) = sumdiv(n, d, (-1)^(bigomega(d) - omega(d)) * eulerphi(n/d)); \\ Michel Marcus, Mar 27 2020
Formula
a(n) = Sum_{k=1..n} (-1)^(bigomega(gcd(n,k)) - omega(gcd(n,k))).
a(n) = Sum_{d|n} mu(n/d) * A327668(d).
From Amiram Eldar, Nov 12 2022: (Start)
Multiplicative with a(p) = p, and a(p^e) = (p^e*(p^2+p-2) - 2*(-1)^e*p)/(p*(p+1)) for e>1.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/5) * Product_{p prime} (1 + 2/p^2) = 0.4381740171... . (End)
Comments