A379109 -id:A379109 - cp's OEIS frontend

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

Antti Karttunen, Dec 17 2024

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

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

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 *)

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));

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)

A379219 Dirichlet inverse of A379218.

Original entry on oeis.org

1, -1, -2, -1, -5, 2, -6, -1, -3, 5, -11, 2, -13, 6, 10, -1, -17, 3, -19, 5, 12, 11, -23, 2, 0, 13, 0, 6, -29, -10, -30, -1, 22, 17, 30, 3, -37, 19, 26, 5, -41, -12, -43, 11, 15, 23, -47, 2, -7, 0, 34, 13, -53, 0, 55, 6, 38, 29, -59, -10, -61, 30, 18, -1, 65, -22, -67, 17, 46, -30, -71, 3, -73, 37, 0, 19, 66, -26, -79, 5

Offset: 1

Antti Karttunen, Dec 18 2024

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

Inverse Möbius transform of A379109.
Dirichlet inverse of A379218.
Cf. A000668, A046692, A336923.

f[p_, e_] := If[2^IntegerExponent[p + 1, 2] == p + 1, Which[e == 1, 1 - p, e == 2, -p, e > 2, 0], If[e == 1, -p, 0]]; f[2, e_] := -1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 03 2025 *)

A046692(n) = { my(f=factor(n)~); prod(i=1, #f, if(1==f[2,i], -(f[1,i]+1), if(2==f[2,i], f[1,i], 0))); };
A209229(n) = (n && !bitand(n,n-1));
A336923(n) = A209229(sigma(n+n)-sigma(n));
A379109(n) = sumdiv(n,d,A046692(d)*A336923(n/d));
A379219(n) = sumdiv(n,d,A379109(d));

a(n) = Sum_{d|n} A379109(d).
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA379218(n/d) * a(d).
Multiplicative with a(2^e) = -1, and for an odd prime p, if p is a Mersenne prime, a(p) = 1-p, a(p^2) = -p, and a(p^e) = 0 for e >= 3, and otherwise a(p) = -p and a(p^e) = 0 for e >= 2. - Amiram Eldar, Jan 03 2025

cp's OEIS Frontend

A379108 Dirichlet convolution of sigma with A359579.

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