A349915 Dirichlet inverse of A113415, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.

1, -1, -3, 0, -4, 3, -5, 0, 1, 4, -7, 0, -8, 5, 10, 0, -10, -1, -11, 0, 12, 7, -13, 0, -1, 8, -1, 0, -16, -10, -17, 0, 16, 10, 14, 0, -20, 11, 18, 0, -22, -12, -23, 0, -2, 13, -25, 0, -5, 1, 22, 0, -28, 1, 18, 0, 24, 16, -31, 0, -32, 17, -2, 0, 20, -16, -35, 0, 28, -14, -37, 0, -38, 20, 5, 0, 20, -18, -41, 0, -2, 22

Offset: 1

Author

Antti Karttunen, Dec 07 2021

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A000593, A001227, A113415, A349916.

Programs

Mathematica

s[n_] := DivisorSum[n, (# + 1) * Mod[#, 2] &] / 2; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Dec 08 2021 *)

PARI

A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
memoA349915 = Map();
A349915(n) = if(1==n,1,my(v); if(mapisdefined(memoA349915,n,&v), v, v = -sumdiv(n,d,if(dA113415(n/d)*A349915(d),0)); mapput(memoA349915,n,v); (v)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A113415(n/d) * a(d).

a(n) = A349916(n) - A113415(n).