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A374188 Array read by ascending antidiagonals: b is a term of row A(a) if and only if K(a/b) != K(A374157(b)/a), where K denotes the Kronecker symbol (A372728), and a = 4*n - 1 for some n >= 1.

%I A374188 #17 Jul 02 2024 06:12:24
%S A374188 2,2,10,2,10,26,2,10,12,28,2,26,12,18,34,2,10,28,18,24,44,2,10,12,34,
%T A374188 24,26,50,2,10,12,18,44,26,34,56,2,10,26,18,24,56,28,44,58,2,12,12,28,
%U A374188 24,26,58,34,48,74,2,10,18,18,34,26,28,74,42,50,76
%N A374188 Array read by ascending antidiagonals: b is a term of row A(a) if and only if  K(a/b) != K(A374157(b)/a), where K denotes the Kronecker symbol (A372728), and a = 4*n - 1 for some n >= 1.
%C A374188 We say two integers, a and b, are related by the golden theorem (Gauss) if K(a/b) = K(A374157(b)/a), an identity, that is valid for all whole numbers a (A001057) and all odd numbers b (A005408). This fact is equivalent to the law of quadratic reciprocity and its first and second supplement. See A372728 (Kronecker) and A373223 (Gauss) for details and examples. Here, we complement this by looking at pairs of integers that do not obey this law.
%F A374188 All terms are even.
%e A374188   [n] [ a] b ...
%e A374188   [1] [ 3] 2, 10, 26, 28, 34, 44, 50, 56, 58, 74, 76, 82, ...  A374180
%e A374188   [2] [ 7] 2, 10, 12, 18, 24, 26, 34, 44, 48, 50, 58, 60, ...  A374181
%e A374188   [3] [11] 2, 10, 12, 18, 24, 26, 28, 34, 42, 48, 50, 56, ...  A374182
%e A374188   [4] [15] 2, 26, 28, 34, 44, 56, 58, 74, 76, 82, 88, 92, ...  A374183
%e A374188   [5] [19] 2, 10, 12, 18, 24, 26, 28, 34, 42, 44, 48, 50, ...  A374184
%e A374188   [6] [23] 2, 10, 12, 18, 24, 26, 28, 34, 42, 44, 48, 50, ...
%e A374188   [7] [27] 2, 10, 26, 28, 34, 44, 50, 56, 58, 74, 76, 82, ...
%e A374188   [8] [31] 2, 10, 12, 18, 24, 26, 28, 34, 42, 44, 48, 50, ...
%p A374188 KS := (a, n) -> NumberTheory:-KroneckerSymbol(a, n):
%p A374188 A374157 := n -> ifelse(iquo(n, 2)::even, n, -n):
%p A374188 A374188_row := (a, len) -> local n; select(n -> (KS(a, n) <> KS(A374157(n), a)), [seq(0..len)]): seq(print(A374188_row(4*m - 1, 350)), m = 1..5);
%o A374188 (SageMath)
%o A374188 def A374157(n): return (-1)**(n // 2)*n
%o A374188 def ks(a, n): return kronecker_symbol(a, n)
%o A374188 def ksp(a, len): return [n for n in range(len) if ks(a, n) != ks(A374157(n), a)]
%o A374188 def A374188_row(n, len): return ksp(4*n - 1, len)
%o A374188 for m in range(1, 8): print(A374188_row(m, 100)[:12])
%Y A374188 Rows: A374180 [1], A374181 [2], A374182 [3], A374183 [4], A374184 [5].
%Y A374188 Cf. A374189 (seen as set), A372728 (Kronecker), A373223 (Gauss), A374157, A004767.
%K A374188 nonn,tabl
%O A374188 1,1
%A A374188 _Peter Luschny_, Jun 30 2024