A349629 -id:A349629 - cp's OEIS frontend

A354365 Numerators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).

1, -2, -3, 0, -5, 3, -7, 0, 0, 10, -11, 0, -13, 14, 5, 0, -17, 0, -19, 0, 21, 22, -23, 0, 0, 26, 0, 0, -29, -5, -31, 0, 33, 34, 7, 0, -37, 38, 39, 0, -41, -21, -43, 0, 0, 46, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 58, -59, 0, -61, 62, 0, 0, 65, -33, -67, 0, 69, -14, -71, 0, -73, 74, 0, 0, 11, -39, -79, 0, 0, 82

Offset: 1

Antti Karttunen, Jun 07 2022

Because the ratio n / A064989(n) = A319626(n) / A319627(n) is multiplicative, so is also its Dirichlet inverse (which also is a sequence of rational numbers). This sequence gives the numerators when presented in its lowest terms, while A354366 gives the denominators. See the examples.

			The ratio a(n)/A354366(n) for n = 1..22: 1, -2, -3/2, 0, -5/3, 3, -7/5, 0, 0, 10/3, -11/7, 0, -13/11, 14/5, 5/2, 0, -17/13, 0, -19/17, 0, 21/10, 22/7.

Index entries for sequences computed from indices in prime factorization

Cf. A013929 (positions of 0's), A055615, A319626, A319627, A354350.
Cf. A354366 (denominators).
Cf. also A349629, A354351, A354827.

A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A354365(n) = numerator((moebius(n)*n)/A064989(n));

a(n) = A055615(n) / gcd(A055615(n), A064989(n)).

A349630 Denominators of the Dirichlet inverse of the abundancy index, sigma(n)/n.

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 1, 3, 5, 11, 3, 13, 7, 5, 1, 17, 2, 19, 5, 21, 11, 23, 1, 5, 13, 1, 7, 29, 5, 31, 1, 11, 17, 35, 6, 37, 19, 39, 1, 41, 7, 43, 11, 5, 23, 47, 1, 7, 10, 17, 13, 53, 1, 55, 1, 57, 29, 59, 5, 61, 31, 21, 1, 65, 11, 67, 17, 23, 35, 71, 1, 73, 37, 15, 19, 77, 13, 79, 1, 1, 41, 83, 21, 85, 43, 29, 1, 89

Offset: 1

Antti Karttunen, Nov 27 2021

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

Cf. A000203, A017665, A017666.

Cf. A349629 (numerators).

f[1] = 1; f[n_] := f[n] = -DivisorSum[n, f[#] * DivisorSigma[1, n/#] * #/n &, # < n &]; Denominator @ Array[f, 100] (* Amiram Eldar, Nov 28 2021 *)

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(dA349630(n) = denominator(vDirInv_of_Abi[n]);

cp's OEIS Frontend

A354365 Numerators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).

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