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

Author

Antti Karttunen, Nov 27 2021

Keywords

• sign
• frac

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000203, A017665, A017666.

Cf. A349630 (denominators).

Programs

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 *)

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(dA349629(n) = numerator(vDirInv_of_Abi[n]);