A379219 Dirichlet inverse of A379218.

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

Author

Antti Karttunen, Dec 18 2024

Keywords

• sign
• mult
• easy

Links

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

Crossrefs

Inverse Möbius transform of A379109.

Dirichlet inverse of A379218.

Cf. A000668, A046692, A336923.

Programs

Mathematica

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 *)

PARI

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));

Formula

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