A325126 a(1) = 1; a(n) = -Sum_{d|n, d<n} rad(n/d) * a(d), where rad = A007947.

1, -2, -3, 2, -5, 6, -7, -2, 6, 10, -11, -6, -13, 14, 15, 2, -17, -12, -19, -10, 21, 22, -23, 6, 20, 26, -12, -14, -29, -30, -31, -2, 33, 34, 35, 12, -37, 38, 39, 10, -41, -42, -43, -22, -30, 46, -47, -6, 42, -40, 51, -26, -53, 24, 55, 14, 57, 58, -59, 30

Offset

1,2

Comments

Dirichlet inverse of A007947.

Moebius transform of A125131.

Links

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

Formula

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} rad(k) * A(x^k).

From Isaac Saffold, May 30 2020: (Start)

a(n) = A008836(n)*A326297(n)*A007947(n).

Proof:

Define lambda(n) := A008836(n); h(n) := A326297(n); rad(n) := A007947(n).

As lambda(n), h(n), and rad(n) are multiplicative, the identity needs only to be proved for prime power n.

It is clear that the identity holds for n = 1 = p^0. For a given nonnegative integer k, assume the identity holds for all v such that 0 <= v <= k. Then, by the recursive formula for Dirichlet inverses,

a(p^(k+1)) = -Sum_{v=0..k} lambda(p^v)*h(p^v)*rad(p^v)*rad(p^(k+1-v))

= -p * (1 + p*Sum_{v=1..k}((-1)^v * (p-1)^(v-1)))

= -p * (1 - p*Sum_{v=0..(k-1)}((1 - p)^v))

= -p * (1 - p*(((1-p)^k - 1) / -p))

= -p * (1-p)^k

= (-1)^(k+1) * (p-1)^k * p

= lambda(p^(k+1)) * h(p^(k+1)) * rad(p^(k+1))

Thus the identity holds for p^(k+1), k >= 0.

As k is arbitrary and the identity holds for p^0, it holds for the prime powers, and thus for all positive integers. Q.E.D. (End)

Mathematica

a[n_] := If[n == 1, n, -Sum[If[d < n, Last[Select[Divisors[n/d], SquareFreeQ]] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 60}]
f[p_, e_] := -p*(1 - p)^(e - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)

Prog

(PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = {my(va=vector(nn)); va[1] = 1; for (n=2, nn, va[n] = -sumdiv(n, d, if (d<n, rad(n/d) * va[d])); ); va; } \\ Michel Marcus, Jun 01 2020

Crossrefs

Cf. A007947, A008836, A125131, A326297.

Cf. also A327564, A332732, A349340, A349612, A349618.

Sequence in context: A243057 A090078 A325814 * A327937 A259445 A080979

Adjacent sequences: A325123 A325124 A325125 * A325127 A325128 A325129

Keyword

sign,mult,easy

Author

Ilya Gutkovskiy, Sep 04 2019

Status

approved