cp's OEIS Frontend

A257403 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).

%I A257403 #23 Oct 17 2022 07:12:07
%S A257403 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,
%T A257403 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,
%U A257403 0,0,0,0,2,0,0,0,0,0,0,0,0,2,2,0,0,2,0
%N A257403 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).
%H A257403 G. C. Greubel, <a href="/A257403/b257403.txt">Table of n, a(n) for n = 1..2500</a>
%F A257403 Moebius transform is the period 288 sequence A257477.
%F A257403 a(3*n) = a(4*n) = a(8*n + 5) = a(8*n + 7) = 0. a(2*n + 1) = a(4*n + 2).
%F A257403 a(6*n + 1) = A257399(n). a(6*n + 5) = 2*A257402(n).
%F A257403 a(24*n + 1) = A257398(n). a(24*n + 11) = 2*A255318(n). a(24*n + 17) = 2*A255319(n). a(24*n + 19) = 2*A255317(n).
%F A257403 From _Michael Somos_, Apr 22 2015: (Start)
%F A257403 a(3*n + 2) = A256505(n) unless n == 5 (mod 8). a(3*n + 19) = 2 * A256574(n) unless n == 2 (mod 8).
%F A257403 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)
%F A257403 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(6*sqrt(2)) = 0.370240... . - _Amiram Eldar_, Oct 17 2022
%e A257403 G.f. = x + x^2 + 2*x^11 + 2*x^17 + 2*x^19 + 2*x^22 + x^25 + 2*x^34 + ...
%t A257403 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])];
%o A257403 (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)))};
%Y A257403 Cf. A255317, A255318, A255319, A256505, A256574, A257398, A257399, A257402, A257477.
%K A257403 nonn,mult
%O A257403 1,11
%A A257403 _Michael Somos_, Apr 21 2015