|
|
A349629
|
|
Numerators of the Dirichlet inverse of the abundancy index, sigma(n)/n.
|
|
3
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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
|
|
|
KEYWORD
|
sign,frac
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|