A349630 -id:A349630 - cp's OEIS frontend

A354366 Denominators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).

1, 1, 2, 1, 3, 1, 5, 1, 1, 3, 7, 1, 11, 5, 2, 1, 13, 1, 17, 1, 10, 7, 19, 1, 1, 11, 1, 1, 23, 1, 29, 1, 14, 13, 3, 1, 31, 17, 22, 1, 37, 5, 41, 1, 1, 19, 43, 1, 1, 1, 26, 1, 47, 1, 21, 1, 34, 23, 53, 1, 59, 29, 1, 1, 33, 7, 61, 1, 38, 3, 67, 1, 71, 31, 1, 1, 5, 11, 73, 1, 1, 37, 79, 1, 39, 41, 46, 1, 83, 1, 55, 1, 58

Offset: 1

Antti Karttunen, Jun 07 2022

Equally, denominators of Dirichlet inverse of fraction n / A064989(n). See also comments in A354365.

Index entries for sequences computed from indices in prime factorization

Cf. A055615, A064989, A319626, A319627, A354360 (positions of 1's).
Cf. A354365 (numerators).
Cf. also A349630.

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); };
A354366(n) = denominator((moebius(n)*n)/A064989(n));

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

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

Original entry on oeis.org

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

Antti Karttunen, Nov 27 2021

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.

			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.

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

Cf. A000203, A017665, A017666.

Cf. A349630 (denominators).

f[1] = 1; f[n_] := f[n] = -DivisorSum[n, f[#] * DivisorSigma[1, n/#] * #/n &, # < n &]; Numerator @ 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(dA349629(n) = numerator(vDirInv_of_Abi[n]);

cp's OEIS Frontend

A354366 Denominators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).

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