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A289741 a(n) = Kronecker symbol (-20/n).

%I A289741 #54 Feb 16 2025 08:33:49
%S A289741 0,1,0,1,0,0,0,1,0,1,0,-1,0,-1,0,0,0,-1,0,-1,0,1,0,1,0,0,0,1,0,1,0,-1,
%T A289741 0,-1,0,0,0,-1,0,-1,0,1,0,1,0,0,0,1,0,1,0,-1,0,-1,0,0,0,-1,0,-1,0,1,0,
%U A289741 1,0,0,0,1,0,1,0,-1,0,-1,0,0,0,-1,0,-1,0
%N A289741 a(n) = Kronecker symbol (-20/n).
%C A289741 Period 20: repeat [0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1].
%C A289741 This sequence is one of the three non-principal real Dirichlet characters modulo 20. The other two are Jacobi or Kronecker symbols {(20/n)} (or {(n/20)}) and {((-100)/n)} (A185276).
%C A289741 Note that (Sum_{i=0..19} i*a(i))/(-20) = 2 gives the class number of the imaginary quadratic field Q(sqrt(-5)). The fact Q(sqrt(-5)) has class number 2 implies that Q(sqrt(-5)) is not a unique factorization domain.
%H A289741 Antti Karttunen, <a href="/A289741/b289741.txt">Table of n, a(n) for n = 0..10000</a>
%H A289741 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KroneckerSymbol.html">Kronecker Symbol</a>
%H A289741 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,-1).
%F A289741 a(n) = 1 for n in A045797; -1 for n in A045798; 0 for n that are not coprime with 20.
%F A289741 Completely multiplicative with a(p) = a(p mod 20) for primes p.
%F A289741 a(n) = A080891(n)*A101455(n).
%F A289741 a(n) = -a(n+10) = -a(-n) for all n in Z.
%F A289741 Multiplicative with a(2) = a(5) = 0, a(p) = (-1)^floor(p/10) otherwise; equivalently: a(n) = (-1)^floor(n/10) if n is coprime to 2*5, 0 otherwise. - _M. F. Hasler_, Feb 28 2022
%t A289741 Array[KroneckerSymbol[-20, #]&, 100, 0] (* _Amiram Eldar_, Jan 10 2019 *)
%o A289741 (PARI) a(n) = kronecker(-20, n)
%Y A289741 Cf. A035170 (inverse Moebius transform).
%Y A289741 Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017 (d=-24), A011586 (d=-23), this sequence (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), A101455 (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), A322829 (d=21), A322796 (d=24).
%Y A289741 Cf. A045797, A045798, A185276.
%K A289741 sign,mult,easy
%O A289741 0,1
%A A289741 _Jianing Song_, Dec 27 2018