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A379108 Dirichlet convolution of sigma with A359579.

%I A379108 #13 Jan 03 2025 05:19:33
%S A379108 1,2,3,4,6,6,7,8,10,12,12,12,14,14,18,16,18,20,20,24,21,24,24,24,31,
%T A379108 28,30,28,30,36,31,32,36,36,42,40,38,40,42,48,42,42,44,48,60,48,48,48,
%U A379108 50,62,54,56,54,60,72,56,60,60,60,72,62,62,70,64,84,72,68,72,72,84,72,80,74,76,93,80,84,84,80,96,91,84,84,84
%N A379108 Dirichlet convolution of sigma with A359579.
%H A379108 Antti Karttunen, <a href="/A379108/b379108.txt">Table of n, a(n) for n = 1..32769</a>
%H A379108 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A379108 a(n) = Sum_{d|n} A000203(d)*A359579(n/d).
%F A379108 From _Amiram Eldar_, Jan 02 2025: (Start)
%F A379108 Multiplicative with a(2^e) = 2^e, and for an odd prime p, a(p^e) = (p^(e+2) + ((-1)^e-1)*(p-1)/2 - 1)/(p^2-1) if p is a Mersenne prime (A000668), and a(p^e) = sigma(p^e) otherwise.
%F A379108 Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/16) / Product_{p in A000668} (1 + 1/p^2) = 0.54346268676686758531... . (End)
%t A379108 f[p_, e_] := If[2^IntegerExponent[p + 1, 2] == p + 1, (p^(e + 2) + ((-1)^e - 1)*(p - 1)/2 - 1)/(p^2 - 1), (p^(e + 1) - 1)/(p - 1)]; f[2, e_] := 2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 02 2025 *)
%o A379108 (PARI)
%o A379108 A209229(n) = (n && !bitand(n,n-1));
%o A379108 A359579(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1], -(1==f[k,2]), (-A209229(1+f[k,1]))^f[k,2])); };
%o A379108 A379108(n) = sumdiv(n,d,sigma(d)*A359579(n/d));
%Y A379108 Cf. A000203, A000668, A054784, A336923, A359579, A379109 (Dirichlet inverse).
%K A379108 nonn,mult
%O A379108 1,2
%A A379108 _Antti Karttunen_, Dec 17 2024