login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A333569 a(n) = Sum_{d|n} (-1)^(bigomega(d) - omega(d)) * phi(n/d). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Moebius transform of A327668.
LINKS
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)
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 *)
PROG
(PARI) a(n) = sumdiv(n, d, (-1)^(bigomega(d) - omega(d)) * eulerphi(n/d)); \\ Michel Marcus, Mar 27 2020
CROSSREFS
Sequence in context: A308056 A336965 A293303 * A110500 A161871 A331737
KEYWORD
nonn,mult
AUTHOR
Ilya Gutkovskiy, Mar 26 2020
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)