A379109 Dirichlet convolution of A046692 (inverse of sigma) with A336923, where A336923(n) = 1 if sigma(2n) - sigma(n) is a power of 2, otherwise 0.

1, -2, -3, 0, -6, 6, -7, 0, -1, 12, -12, 0, -14, 14, 18, 0, -18, 2, -20, 0, 21, 24, -24, 0, 5, 28, 3, 0, -30, -36, -31, 0, 36, 36, 42, 0, -38, 40, 42, 0, -42, -42, -44, 0, 6, 48, -48, 0, -1, -10, 54, 0, -54, -6, 72, 0, 60, 60, -60, 0, -62, 62, 7, 0, 84, -72, -68, 0, 72, -84, -72, 0, -74, 76, -15, 0, 84, -84, -80, 0, 0, 84

Offset: 1

Author

Antti Karttunen, Dec 17 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

Cf. A000203, A000668, A046692, A054784, A336923, A379108 (Dirichlet inverse).

Programs

Mathematica

f[p_, e_] := If[2^IntegerExponent[p + 1, 2] == p + 1, Which[e == 1, -p, e == 2, -1, e == 3, p, e > 3, 0], Which[e == 1, -p - 1, e == 2, p, e > 2, 0]]; f[2, e_] := If[e == 1, -2, 0]; 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));

Formula

a(n) = Sum_{d|n} A046692(d)*A336923(n/d).

Multiplicative with a(2^e) = -2 if e = 1 and 0 otherwise, and for an odd prime p, if p is a Mersenne prime, a(p) = -p, a(p^2) = -1, a(p^3) = p, and a(p^e) = 0 for e >= 4, and otherwise a(p) = -(p+1), a(p^2) = p and a(p^e) = 0 for e >= 3. - Amiram Eldar, Jan 03 2025