%I #14 Nov 12 2022 05:34:46

%S 1,1,2,3,4,2,6,9,8,4,10,6,12,6,8,27,16,8,18,12,12,10,22,18,24,12,32,

%T 18,28,8,30,81,20,16,24,24,36,18,24,36,40,12,42,30,32,22,46,54,48,24,

%U 32,36,52,32,40,54,36,28,58,24,60,30,48,243,48,20,66,48,44,24,70,72,72,36,48,54,60,24,78,108,128,40

%N Multiplicative with a(p^e) = (p - 1) * (p + 1)^(e-1).

%H Antti Karttunen, <a href="/A340368/b340368.txt">Table of n, a(n) for n = 1..16383</a>

%F a(n) = A167344(n) / A340323(n).

%F a(n) = A173557(n) * A327564(n).

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 3/p^2 + 2/p^4) / Product_{p prime} (1 - 2/p^2 - 1/p^3) = 0.4313799748... . - _Amiram Eldar_, Nov 12 2022

%t f[p_, e_] := (p - 1)*(p + 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 12 2022 *)

%o (PARI) A340368(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,(f[i,1]-1)*((f[i,1]+1)^(f[i,2]-1))));

%Y Cf. A167344, A173557, A327564, A340323.

%K nonn,mult

%O 1,3

%A _Antti Karttunen_, Jan 06 2021