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

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..32769

Index entries for sequences related to sigma(n).

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)

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 *)

Prog

(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));

Crossrefs

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

Sequence in context: A106006 A364104 A050460 * A246594 A175808 A334666

Adjacent sequences: A379105 A379106 A379107 * A379109 A379110 A379111

Keyword

nonn,mult

Author

Antti Karttunen, Dec 17 2024

Status

approved