A062953 Multiplicative with a(p^e) = -p.

1, -2, -3, -2, -5, 6, -7, -2, -3, 10, -11, 6, -13, 14, 15, -2, -17, 6, -19, 10, 21, 22, -23, 6, -5, 26, -3, 14, -29, -30, -31, -2, 33, 34, 35, 6, -37, 38, 39, 10, -41, -42, -43, 22, 15, 46, -47, 6, -7, 10, 51, 26, -53, 6, 55, 14, 57, 58, -59, -30, -61, 62, 21, -2, 65, -66, -67, 34, 69, -70, -71, 6, -73, 74, 15, 38, 77, -78, -79

Offset

1,2

Comments

Except for first term, row products of A142971. - Mats Granvik and Gary W. Adamson, Jul 15 2008

Dirichlet inverse of A003968. - Werner Schulte, Oct 25 2018

Links

Ivan Neretin, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|n} mu(d)*sigma(d) = Sum_{d|n} A063441(d).

From Enrique Pérez Herrero, Aug 24 2010: (Start)

a(n) = Sum_{d|n} mu(d)*psi(d), where psi is A001615.

a(n) = rad(n)*(-1)^omega(n) = A007947(n)*(-1)^A001221(n). (End)

G.f.: Sum_{k>=1} mu(k)*sigma(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Feb 19 2017

a(n) = (n*invphi(n))/phi(n) = (n*A023900(n))/(A000010(n)). - Amrit Awasthi, Jan 30 2022

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 - 1/p^s - 1/p^(s-1)). - Amiram Eldar, Sep 18 2023

Maple

with(numtheory): seq(coeff(series(add(mobius(k)*sigma(k)*x^k/(1-x^k), k=1..n), x, n+1), x, n), n = 1 .. 80); # Muniru A Asiru, Oct 26 2018

Mathematica

A062953[n_]:=DivisorSum[n, MoebiusMu[ # ]*DivisorSigma[1, # ]&] (* Enrique Pérez Herrero, Aug 24 2010 *)

Prog

(PARI) a(n) = sumdiv(n, d, moebius(d)*sigma(d)); \\ Michel Marcus, Feb 19 2017

Crossrefs

Apart from signs, essentially same as A007947.

Cf. A001221, A001615, A003968, A063441, A142971.

Cf. A000010, A023900.

Sequence in context: A088835 A007947 A015053 * A347230 A015052 A348036

Adjacent sequences: A062950 A062951 A062952 * A062954 A062955 A062956

Keyword

mult,sign,easy

Author

Vladeta Jovovic, Jul 21 2001

Status

approved