A101035 -id:A101035 - cp's OEIS frontend

A351403 G.f. A(x) satisfies: (1 - x) = Product_{i>=1, j>=1} A(x^(i*j)).

1, -1, 2, 0, 0, 4, -5, 9, -6, 3, 4, -9, 15, -17, 13, -8, 0, 1, -9, 12, -17, 15, -25, 29, -27, 12, -3, -14, 28, -55, 63, -54, 53, -46, 18, 32, -57, 85, -106, 122, -108, 43, 8, -29, 80, -161, 148, -115, 104, -78, 57, 29, -77, 89, -99, 263, -283, 182, -212, 133, 49

Offset: 0

Ilya Gutkovskiy, Feb 10 2022

sign

Convolution inverse of A351402.

Cf. A000005, A007427, A101035, A117208, A288098, A351402.

nmax = 60; A007427[n_] := Sum[MoebiusMu[d] MoebiusMu[n/d], {d, Divisors[n]}]; CoefficientList[Series[Product[(1 - x^k)^A007427[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f. A(x) satisfies: (1 - x) = Product_{k>=1} A(x^k)^A000005(k).
G.f.: Product_{k>=1} (1 - x^k)^A007427(k).
G.f.: exp( -Sum_{k>=1} A101035(k) * x^k / k ).
a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A101035(k) * a(n-k).

A351654 Dirichlet g.f.: zeta(s) / (zeta(s-1) * zeta(s-2)).

Original entry on oeis.org

1, -5, -11, 3, -29, 55, -55, 3, 16, 145, -131, -33, -181, 275, 319, 3, -305, -80, -379, -87, 605, 655, -551, -33, 96, 905, 16, -165, -869, -1595, -991, 3, 1441, 1525, 1595, 48, -1405, 1895, 1991, -87, -1721, -3025, -1891, -393, -464, 2755, -2255, -33, 288, -480, 3355, -543, -2861, -80, 3799

Offset: 1

Ilya Gutkovskiy, Feb 16 2022

Dirichlet inverse of A069097.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A023900, A046970, A055615, A069097, A101035, A328254, A334657.

A069097[n_] := Sum[GCD[n, k]^2, {k, 1, n}]; a[1] = 1; a[n_] := a[n] = -Sum[A069097[n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 55}]
f[p_, e_] := If[e == 1, 0, p^3] - p^2 - p + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 25 2025 *)

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA069097(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d));
v351654 = DirInverseCorrect(vector(up_to, n, A069097(n)));
A351654(n) = v351654[n]; \\ Antti Karttunen, Feb 16 2022

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X)*(1 - p^2*X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Feb 16 2022

a(1) = 1; a(n) = -Sum_{d|n, d < n} A069097(n/d) * a(d).
a(n) = Sum_{d|n} A023900(n/d) * A334657(d).
a(n) = Sum_{d|n} A046970(n/d) * A055615(d).
a(n) = Sum_{d|n} A000005(n/d) * A328254(d).
Multiplicative with a(p) = -p^2 - p + 1, and a(p^e) = p^3 - p^2 - p + 1 for e >= 2. - Amiram Eldar, May 25 2025

cp's OEIS Frontend

A351403 G.f. A(x) satisfies: (1 - x) = Product_{i>=1, j>=1} A(x^(i*j)).

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