A379108 Dirichlet convolution of sigma with A359579.

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, 28, 30, 28, 30, 36, 31, 32, 36, 36, 42, 40, 38, 40, 42, 48, 42, 42, 44, 48, 60, 48, 48, 48, 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

Offset: 1

Author

Antti Karttunen, Dec 17 2024

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..32769
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A000668, A054784, A336923, A359579, A379109 (Dirichlet inverse).

Programs

Mathematica

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 *)

PARI

A209229(n) = (n && !bitand(n,n-1));
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])); };
A379108(n) = sumdiv(n,d,sigma(d)*A359579(n/d));

Formula

a(n) = Sum_{d|n} A000203(d)*A359579(n/d).

From Amiram Eldar, Jan 02 2025: (Start)

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.

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)