A379218 Möbius transform of A379108.

1, 1, 2, 2, 5, 2, 6, 4, 7, 5, 11, 4, 13, 6, 10, 8, 17, 7, 19, 10, 12, 11, 23, 8, 25, 13, 20, 12, 29, 10, 30, 16, 22, 17, 30, 14, 37, 19, 26, 20, 41, 12, 43, 22, 35, 23, 47, 16, 43, 25, 34, 26, 53, 20, 55, 24, 38, 29, 59, 20, 61, 30, 42, 32, 65, 22, 67, 34, 46, 30, 71, 28, 73, 37, 50, 38, 66, 26, 79, 40, 61, 41, 83, 24

Offset

1,3

Comments

Dirichlet convolution of A000027 with A359579.

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} d*A359579(n/d).

a(n) = Sum_{d|n} A008683(d)*A379108(n/d).

From Amiram Eldar, Jan 02 2025: (Start)

Multiplicative with a(2^e) = 2^(e-1), and for an odd prime p, a(p^e) = (p^(e + 1) + (-1)^e)/(p + 1) if p is a Mersenne prime (A000668), and a(p^e) = p^e otherwise.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (3/8) / Product_{p in A000668} (1 + 1/p^2) = 0.33038569613198448017... . (End)

Mathematica

f[p_, e_] := If[2^IntegerExponent[p + 1, 2] == p + 1, (p^(e + 1) + (-1)^e)/(p + 1), p^e]; f[2, e_] := 2^(e - 1); 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])); };
A379218(n) = sumdiv(n, d, d*A359579(n/d));

Crossrefs

Cf. A000027, A000203, A000668, A008683, A336923, A359579, A379108, A379219 (Dirichlet inverse).

Cf. also A026741.

Sequence in context: A018216 A059907 A359101 * A383180 A024931 A256612

Adjacent sequences: A379215 A379216 A379217 * A379219 A379220 A379221

Keyword

nonn,mult

Author

Antti Karttunen, Dec 18 2024

Status

approved