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%I #21 Oct 24 2023 04:37:11
%S 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,
%T 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,
%U 1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0
%N a(n) = 1 if n not == 2 (mod 4) and relatively prime to 63 otherwise 0.
%H G. C. Greubel, <a href="/A209198/b209198.txt">Table of n, a(n) for n = 1..5000</a>
%F G.f.: B(x) - B(x^7) - B(x^9) + B(x^63) where B(x) = (x + x^3 + x^4) / (1 - x^4).
%F 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.
%F a(n) = a(n + 252) = a(-n), a(4*n + 2) = a(7*n) = a(9*n) = 0 for all n in Z.
%F Euler transform is A119952.
%F From _Amiram Eldar_, Oct 24 2023: (Start)
%F Dirichlet g.f.: zeta(s) * (1 - 1/2^s + 1/4^s) * (1 - 1/9^s) * (1 - 1/7^s).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4/7. (End)
%e G.f. = x + x^3 + x^4 + x^5 + x^8 + x^11 + x^12 + x^13 + x^15 + x^16 + x^17 + ...
%t 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 *)
%o (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)))};
%o (PARI) {a(n) = b(n) - b(n/7) - b(n/9) + b(n/63)}; {b(n) = if( n == round(n), n%4 != 2, 0)};
%o (PARI) {a(n) = n%4!=2 && n%7!=0 && n%9!=0};
%Y Cf. A119952.
%K nonn,easy,mult
%O 1,1
%A _Michael Somos_, Mar 05 2012
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