A257477 Multiplicative with a(2) = 0, a(4) = -1, a(2^e) = 0 if e>2, a(3) = -1, a(3^e) = 0^e if e>1, a(p^e) = 1 if p == 1, 3 (mod 8), a(p^e) = (-1)^e if p == 5, 7 (mod 8).

1, 0, -1, -1, -1, 0, -1, 0, 0, 0, 1, 1, -1, 0, 1, 0, 1, 0, 1, 1, 1, 0, -1, 0, 1, 0, 0, 1, -1, 0, -1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 1, 0, 1, -1, 0, 0, -1, 0, 1, 0, -1, 1, -1, 0, -1, 0, -1, 0, 1, -1, -1, 0, 0, 0, 1, 0, 1, -1, 1, 0, -1, 0, 1, 0, -1, -1, -1, 0, -1

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..2500

Formula

G.f.: f(x) - 2*f(x^3) - f(x^4) + f(x^9) + 2*f(x^12) - f(x^36) where f(x) = (x + x^3) / (1 + x^4) is the g.f. for A188510.

abs(a(2*n + 1)) = A168182(n+5).

a(4*n + 2) = a(8*n) = a(9*n) = 0.

a(n) = -a(-n) = a(n + 288) for all n in Z.

Moebius transform of A257403.

Sum_{k=1..n} abs(a(k)) ~ 5*n/9. - Amiram Eldar, Jan 29 2024

Example

G.f. = x - x^3 - x^4 - x^5 - x^7 + x^11 + x^12 - x^13 + x^15 + x^17 + ...

Mathematica

a[ n_] := Sign[n] If[ Abs[n] < 2, 1, Times @@ (Which[ # < 5, -Boole[# + #2 == 4], Mod[#, 8] < 4, 1, True, (-1)^#2] & @@@ FactorInteger[Abs@n])];
f[x_] := (x + x^3)/(1 + x^4); CoefficientList[Series[f[x] - 2*f[x^3] - f[x^4] + f[x^9] + 2*f[x^12] - f[x^36], {x, 0, 50}], x] (* G. C. Greubel, Aug 03 2018 *)

Prog

(PARI) {a(n) = my(A, p, e); if( n==0, 0, A = factor(abs(n)); sign(n) * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, -(p+e==4), if( p%8 < 4, 1, (-1)^e))))};

Crossrefs

Cf. A168182, A188510, A257403.

Sequence in context: A267053 A359591 A359592 * A259024 A323045 A104106

Adjacent sequences: A257474 A257475 A257476 * A257478 A257479 A257480

Keyword

sign,mult

Author

Michael Somos, Apr 25 2015

Extensions

Definition corrected by Georg Fischer, Jul 23 2022

Status

approved