A327268 -id:A327268 - cp's OEIS frontend

A332793 a(1) = 1; a(n) = n * Sum_{d|n, d < n} (-1)^(n/d) * a(d) / d.

1, 2, -3, 8, -5, -6, -7, 32, 0, -10, -11, -24, -13, -14, 15, 128, -17, 0, -19, -40, 21, -22, -23, -96, 0, -26, 0, -56, -29, 30, -31, 512, 33, -34, 35, 0, -37, -38, 39, -160, -41, 42, -43, -88, 0, -46, -47, -384, 0, 0, 51, -104, -53, 0, 55, -224, 57, -58, -59, 120

Offset: 1

Ilya Gutkovskiy, Feb 24 2020

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

Cf. A002129, A038838 (positions of 0's), A055615, A067856, A327268, A361987.
Partial sums give A361982.
Dirichlet inverse of A181983.

a[1] = 1; a[n_] := n Sum[If[d < n, (-1)^(n/d) a[d]/d, 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]
terms = 60; A[] = 0; Do[A[x] = x + Sum[(-1)^k k A[x^k], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest
f[p_, e_] := If[p == 2, p^(2*e - 1), -p*Boole[e == 1]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} (-1)^k * k * A(x^k).
Dirichlet g.f.: 1 / (zeta(s-1) * (1 - 2^(2 - s))).
a(n) = Sum_{d|n} A327268(d).
Multiplicative with a(2^e) = 2^(2*e-1), and a(p^e) = -p if e=1 and 0 for e>1, for odd primes p. - Amiram Eldar, Dec 02 2020

A327274 Dirichlet g.f.: 1 / (zeta(s)^2 * (1 - 2^(1 - s))).

Original entry on oeis.org

1, 0, -2, 1, -2, 0, -2, 2, 1, 0, -2, -2, -2, 0, 4, 4, -2, 0, -2, -2, 4, 0, -2, -4, 1, 0, 0, -2, -2, 0, -2, 8, 4, 0, 4, 1, -2, 0, 4, -4, -2, 0, -2, -2, -2, 0, -2, -8, 1, 0, 4, -2, -2, 0, 4, -4, 4, 0, -2, 4, -2, 0, -2, 16, 4, 0, -2, -2, 4, 0, -2, 2, -2, 0, -2, -2, 4, 0, -2, -8

Offset: 1

Ilya Gutkovskiy, Oct 22 2019

Dirichlet inverse of A048272.

Moebius transform of A067856.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007427, A008683, A048272, A062503 (positions of 1's), A067856, A327268.

a[1] = 1; a[n_] := Sum[Sum[(-1)^j, {j, Divisors[n/d]}] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 80}]
f[p_, e_] := Switch[e, 1, -2, 2, 1, , 0]; f[2, e] := 2^(e-2); f[2, 1] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

A067856(n) = { my(k); if(n<1, 0, k=valuation(n, 2); moebius(n/2^k)*2^max(0, k-1)); }; \\ From A067856
A327274(n) = sumdiv(n,d,moebius(n/d)*A067856(d));

a(1) = 1; a(n) = -Sum_{d|n, dA048272(n/d) * a(d).
a(n) = Sum_{d|n} mu(n/d) * A067856(d).
a(n) = 0 if n == 2 (mod 4). - Bernard Schott, Dec 07 2021
Multiplicative with a(2) = 0, a(2^e) = 2^(e-2) for e >= 2, and for an odd prime p, a(p) = -2, a(p^2) = 1, and a(p^e) = 0 for e >= 3. - Amiram Eldar, Sep 15 2023

cp's OEIS Frontend

A332793 a(1) = 1; a(n) = n * Sum_{d|n, d < n} (-1)^(n/d) * a(d) / d.

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