OFFSET
1,2
COMMENTS
Absolute values of the Möbius transform of A055491. (See Benoit Cloitre's May 31 2002 comment in A046970).
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p^e) = p^2 - 1.
a(n) = Product_{p prime divides n} (p^2 - 1).
a(n) = abs(A046970(n)).
G.f. for a signed version of the sequence: Sum_{n >= 1} mu(n)*n^2*x^n/(1 - x^n) = Sum_{n >= 1} (-1)^omega(n)*a(n)*x^n = x - 3*x^2 - 8*x^3 - 3*x^4 - 24*x^5 + 24*x^6 - 48*x^7 - 3*x^8 - 8*x^9 + 72*x^10 - ..., where mu(n) is the Möbius function A008683(n) and omega(n) = A001221(n) is the number of distinct primes dividing n. - Peter Bala, Mar 05 2022
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (p-1)*(p^2 + 2*p + 2)/(p*(p^2 + p + 1)) = 0.187556464... . - Amiram Eldar, Oct 22 2022
MAPLE
a:= n-> mul(i[1]^2-1, i=ifactors(n)[2]):
seq(a(n), n=1..80); # Alois P. Heinz, Jan 05 2021
MATHEMATICA
a[n_] := If[n==1, 1, Times @@ ((#^2-1)& @@@ FactorInteger[n])]; Array[a, 50] (* Amiram Eldar, Dec 05 2018 *)
PROG
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Antti Karttunen, Dec 04 2018
STATUS
approved