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A257403
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Multiplicative with a(2) = 1, a(2^e) = 0 if e>1, a(3^e) = 0^e, a(p^e) = e+1 if p == 1, 3 (mod 8), a(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8).
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4
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1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0
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OFFSET
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1,11
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LINKS
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FORMULA
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Moebius transform is the period 288 sequence A257477.
a(3*n) = a(4*n) = a(8*n + 5) = a(8*n + 7) = 0. a(2*n + 1) = a(4*n + 2).
a(3*n + 2) = A256505(n) unless n == 5 (mod 8). a(3*n + 19) = 2 * A256574(n) unless n == 2 (mod 8).
Expansion of F(q) + F(q^2) + G(q) + G(q^2) in powers of q where F(q) = q * A257399(q^6) and G(q) = 2 * q^11 * A257402(q^6). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(6*sqrt(2)) = 0.370240... . - Amiram Eldar, Oct 17 2022
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EXAMPLE
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G.f. = x + x^2 + 2*x^11 + 2*x^17 + 2*x^19 + 2*x^22 + x^25 + 2*x^34 + ...
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MATHEMATICA
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a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (Which[ # == 2, Boole[#2 == 1], # == 3, 0, Mod[#, 8] < 4, #2 + 1, True, Mod[#2 + 1, 2]]& @@@ FactorInteger[n])];
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PROG
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(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p+e==3, p%8 > 4, 1-e%2, e+1)))};
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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