A349432 Dirichlet convolution of A000027 (the identity function) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

1, 1, 1, 2, 2, 1, 3, 4, 2, 2, 5, 2, 6, 3, 0, 8, 8, 2, 9, 4, 0, 5, 11, 4, 6, 6, 4, 6, 14, 0, 15, 16, 0, 8, 0, 4, 18, 9, 0, 8, 20, 0, 21, 10, -2, 11, 23, 8, 12, 6, 0, 12, 26, 4, 0, 12, 0, 14, 29, 0, 30, 15, -3, 32, 0, 0, 33, 16, 0, 0, 35, 8, 36, 18, -4, 18, 0, 0, 39, 16, 8, 20, 41, 0, 0, 21, 0, 20, 44, -2, 0, 22, 0, 23

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Mathematica

k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; kinv[1] = 1; kinv[n_] := kinv[n] = -DivisorSum[n, kinv[#] * k[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * kinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

Prog

(PARI)
up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
A003602(n) = (1+(n>>valuation(n, 2)))/2;
v349134 = DirInverseCorrect(vector(up_to, n, A003602(n)));
A349134(n) = v349134[n];
A003602(n) = (1+(n>>valuation(n, 2)))/2;
A055615(n) = (n*moebius(n));
A349432(n) = sumdiv(n, d, d*A349134(n/d));

Crossrefs

Cf. A003602, A055615, A349134, A349431 (Dirichlet inverse), A349433 (sum with it).

Cf. also A349445, A349448.

Sequence in context: A035317 A368296 A103923 * A186711 A061987 A323899

Adjacent sequences: A349429 A349430 A349431 * A349433 A349434 A349435

Keyword

sign

Author

Antti Karttunen, Nov 17 2021

Status

approved