A349629 Numerators of the Dirichlet inverse of the abundancy index, sigma(n)/n.

1, -3, -4, 1, -6, 2, -8, 0, 1, 9, -12, -2, -14, 12, 8, 0, -18, -1, -20, -3, 32, 18, -24, 0, 1, 21, 0, -4, -30, -12, -32, 0, 16, 27, 48, 1, -38, 30, 56, 0, -42, -16, -44, -6, -2, 36, -48, 0, 1, -3, 24, -7, -54, 0, 72, 0, 80, 45, -60, 4, -62, 48, -8, 0, 84, -24, -68, -9, 32, -72, -72, 0, -74, 57, -4, -10, 96, -28

Offset

1,2

Comments

Because the ratio A000203(n)/n [known as the abundancy index of n] is multiplicative, so is also its Dirichlet inverse. This sequence gives the numerator of that ratio when presented in its lowest terms, while A349630 gives the denominators. See the examples.

Links

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

Example

The ratio a(n)/A349630(n) for n = 1..15: 1/1, -3/2, -4/3, 1/2, -6/5, 2/1, -8/7, 0/1, 1/3, 9/5, -12/11, -2/3, -14/13, 12/7, 8/5.

Mathematica

f[1] = 1; f[n_] := f[n] = -DivisorSum[n, f[#] * DivisorSigma[1, n/#] * #/n &, # < n &]; Numerator @ Array[f, 100] (* Amiram Eldar, Nov 28 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.
Abi(n) = (sigma(n)/n);
vDirInv_of_Abi = DirInverseCorrect(vector(up_to, n, Abi(n)));
A349629(n) = numerator(vDirInv_of_Abi[n]);

Crossrefs

Cf. A000203, A017665, A017666.

Cf. A349630 (denominators).

Sequence in context: A276433 A343226 A030707 * A254525 A132179 A378210

Adjacent sequences: A349626 A349627 A349628 * A349630 A349631 A349632

Keyword

sign,frac

Author

Antti Karttunen, Nov 27 2021

Status

approved