A256232 Multiplicative with a(2^e) = 1-e, a(3^e) = 1, a(p^e) = e+1 if p>3.

1, 0, 1, -1, 2, 0, 2, -2, 1, 0, 2, -1, 2, 0, 2, -3, 2, 0, 2, -2, 2, 0, 2, -2, 3, 0, 1, -2, 2, 0, 2, -4, 2, 0, 4, -1, 2, 0, 2, -4, 2, 0, 2, -2, 2, 0, 2, -3, 3, 0, 2, -2, 2, 0, 4, -4, 2, 0, 2, -2, 2, 0, 2, -5, 4, 0, 2, -2, 2, 0, 2, -2, 2, 0, 3, -2, 4, 0, 2, -6

Offset

1,5

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

Moebius transform is period 6 sequence [1, -1, 0, -1, 1, 0, ...].

G.f.: Sum_{k>0} x^k / (1 + x^k) * Kronecker(9, k).

G.f.: Sum_{k>0} x^k / (1 - x^k) * (-1)^floor(k/3) * Kronecker(-3, k).

G.f.: Sum_{k>0} x^k / (1 - (-x)^k) * (-1)^(k mod 4 = 0) * (k mod 3 > 0).

G.f.: Sum_{k>0} (x^k - x^(2*k)) / (1 + x^(3*k)).

G.f.: Sum_{k>0} (x^k + x^(2*k)) / (1 - x^(3*k)) * (-1)^(k-1).

a(n) = (-1)^(n mod 4 = 0) * A099751(n).

From Amiram Eldar, Nov 30 2022: (Start)

Dirichlet g.f.: zeta(s)^2*(1 - 2^(1-s))*(1 - 1/3^s).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(2)/3 = 0.462098... . (End)

Example

G.f. = x + x^3 - x^4 + 2*x^5 + 2*x^7 - 2*x^8 + x^9 + 2*x^11 - x^12 + ...

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -3, d] (-1)^Quotient[ d, 3], {d, Divisors@n}]];
a[ n_] := SeriesCoefficient[ Sum[ (x^k - x^(2*k)) / (1 + x^(3*k)), {k, n}], {x, 0, n}];

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(d\3) * kronecker( -3, d)))};
(Haskell)
a099751 n = product $ zipWith f (a027748_row n) (a124010_row n)
   where f 2 e = e - 1; f 3 e = 1; f _ e = e + 1
-- Reinhard Zumkeller, Mar 20 2015

Crossrefs

Cf. A099751.

Cf. A027748, A124010.

Sequence in context: A082115 A161553 A366444 * A099751 A159937 A058728

Adjacent sequences: A256229 A256230 A256231 * A256233 A256234 A256235

Keyword

sign,mult

Author

Michael Somos, Mar 19 2015

Status

approved