A209198 a(n) = 1 if n not == 2 (mod 4) and relatively prime to 63 otherwise 0.

1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Formula

G.f.: B(x) - B(x^7) - B(x^9) + B(x^63) where B(x) = (x + x^3 + x^4) / (1 - x^4).

a(n) is multiplicative with a(2^e) = 1 if e!=1, a(3^e) = 1 if e<2, a(7^e) = 0^e, a(p^e) = 1 otherwise.

a(n) = a(n + 252) = a(-n), a(4*n + 2) = a(7*n) = a(9*n) = 0 for all n in Z.

Euler transform is A119952.

From Amiram Eldar, Oct 24 2023: (Start)

Dirichlet g.f.: zeta(s) * (1 - 1/2^s + 1/4^s) * (1 - 1/9^s) * (1 - 1/7^s).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4/7. (End)

Example

G.f. = x + x^3 + x^4 + x^5 + x^8 + x^11 + x^12 + x^13 + x^15 + x^16 + x^17 + ...

Mathematica

B[x_]:= (x + x^3 + x^4)/(1 - x^4); CoefficientList[Series[B[x] - B[x^7] - B[x^9] + B[x^63], {x, 0, 50}], x] (* G. C. Greubel, Aug 11 2018 *)

Prog

(PARI) {a(n) = my(A, p, e); n = abs(n); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, e>1, p==3, e<2,  p==7, 0, 1)))};
(PARI) {a(n) = b(n) - b(n/7) - b(n/9) + b(n/63)}; {b(n) = if( n == round(n), n%4 != 2, 0)};
(PARI) {a(n) = n%4!=2 && n%7!=0 && n%9!=0};

Crossrefs

Cf. A119952.

Sequence in context: A230298 A000480 A118251 * A282343 A099076 A282339

Adjacent sequences: A209195 A209196 A209197 * A209199 A209200 A209201

Keyword

nonn,easy,mult

Author

Michael Somos, Mar 05 2012

Status

approved