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A322360 Multiplicative with a(p^e) = p^2 - 1.

%I A322360 #35 Oct 31 2024 01:52:47
%S A322360 1,3,8,3,24,24,48,3,8,72,120,24,168,144,192,3,288,24,360,72,384,360,
%T A322360 528,24,24,504,8,144,840,576,960,3,960,864,1152,24,1368,1080,1344,72,
%U A322360 1680,1152,1848,360,192,1584,2208,24,48,72,2304,504,2808,24,2880,144,2880,2520,3480,576,3720,2880,384,3,4032,2880
%N A322360 Multiplicative with a(p^e) = p^2 - 1.
%C A322360 Absolute values of A046970, the Dirichlet inverse of the Jordan function J_2 (A007434).
%C A322360 Absolute values of the Möbius transform of A055491. (See _Benoit Cloitre_'s May 31 2002 comment in A046970).
%H A322360 Antti Karttunen, <a href="/A322360/b322360.txt">Table of n, a(n) for n = 1..10000</a>
%F A322360 Multiplicative with a(p^e) = p^2 - 1.
%F A322360 a(n) = Product_{p prime divides n} (p^2 - 1).
%F A322360 a(n) = abs(A046970(n)).
%F A322360 a(n) = A048250(n) * A173557(n) = A066086(n) * A322359(n).
%F A322360 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
%F A322360 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
%F A322360 a(n) = A007434(A007947(n)). - _Enrique Pérez Herrero_, Oct 14 2024
%p A322360 a:= n-> mul(i[1]^2-1, i=ifactors(n)[2]):
%p A322360 seq(a(n), n=1..80);  # _Alois P. Heinz_, Jan 05 2021
%t A322360 a[n_] := If[n==1, 1, Times @@ ((#^2-1)& @@@ FactorInteger[n])]; Array[a, 50]  (* _Amiram Eldar_, Dec 05 2018 *)
%o A322360 (PARI) A322360(n) = factorback(apply(p -> (p*p)-1, factor(n)[, 1]));
%o A322360 (PARI) A322360(n) = abs(sumdiv(n,d,moebius(n/d)*(core(d)^2)));
%Y A322360 Absolute values of A046970.
%Y A322360 Cf. A048250, A173557, A066086, A322359, A351265.
%K A322360 nonn,mult
%O A322360 1,2
%A A322360 _Antti Karttunen_, Dec 04 2018