A349916 Sum of A113415 and its Dirichlet inverse, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.

2, 0, 0, 1, 0, 6, 0, 1, 9, 8, 0, 3, 0, 10, 24, 1, 0, 7, 0, 4, 30, 14, 0, 3, 16, 16, 21, 5, 0, 4, 0, 1, 42, 20, 40, 8, 0, 22, 48, 4, 0, 6, 0, 7, 40, 26, 0, 3, 25, 18, 60, 8, 0, 23, 56, 5, 66, 32, 0, 14, 0, 34, 53, 1, 64, 10, 0, 10, 78, 12, 0, 8, 0, 40, 70, 11, 70, 12, 0, 4, 61, 44, 0, 18, 80, 46, 96, 7, 0, 44, 80

Offset

1,1

Links

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

Formula

a(n) = A113415(n) + A349915(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A113415(d) * A349915(n/d).

For all n >= 1, a(4*n) = A113415(n).

Mathematica

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

Prog

(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(d<n, A113415(n/d)*A349915(d), 0)); mapput(memoA349915, n, v); (v)));
A349916(n) = (A113415(n)+A349915(n));

Crossrefs

Cf. A113415 (also a quadrisection of this sequence), A349915.

Cf. also A349913, A349914.

Sequence in context: A335156 A158785 A346243 * A349342 A365712 A349914

Adjacent sequences: A349913 A349914 A349915 * A349917 A349918 A349919

Keyword

nonn

Author

Antti Karttunen, Dec 07 2021

Status

approved