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.

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, 24, 26, 50, 2, 10, 12, 18, 44, 26, 34, 56, 2, 10, 26, 18, 24, 56, 28, 44, 58, 2, 12, 12, 28, 24, 26, 58, 34, 48, 74, 2, 10, 18, 18, 34, 26, 28, 74, 42, 50, 76

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..66.

Formula

All terms are even.

Example

  [n] [ a] b ...
  [1] [ 3] 2, 10, 26, 28, 34, 44, 50, 56, 58, 74, 76, 82, ...  A374180
  [2] [ 7] 2, 10, 12, 18, 24, 26, 34, 44, 48, 50, 58, 60, ...  A374181
  [3] [11] 2, 10, 12, 18, 24, 26, 28, 34, 42, 48, 50, 56, ...  A374182
  [4] [15] 2, 26, 28, 34, 44, 56, 58, 74, 76, 82, 88, 92, ...  A374183
  [5] [19] 2, 10, 12, 18, 24, 26, 28, 34, 42, 44, 48, 50, ...  A374184
  [6] [23] 2, 10, 12, 18, 24, 26, 28, 34, 42, 44, 48, 50, ...
  [7] [27] 2, 10, 26, 28, 34, 44, 50, 56, 58, 74, 76, 82, ...
  [8] [31] 2, 10, 12, 18, 24, 26, 28, 34, 42, 44, 48, 50, ...

Maple

KS := (a, n) -> NumberTheory:-KroneckerSymbol(a, n):
A374157 := n -> ifelse(iquo(n, 2)::even, n, -n):
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);

Prog

(SageMath)
def A374157(n): return (-1)**(n // 2)*n
def ks(a, n): return kronecker_symbol(a, n)
def ksp(a, len): return [n for n in range(len) if ks(a, n) != ks(A374157(n), a)]
def A374188_row(n, len): return ksp(4*n - 1, len)
for m in range(1, 8): print(A374188_row(m, 100)[:12])

Crossrefs

Rows: A374180 [1], A374181 [2], A374182 [3], A374183 [4], A374184 [5].

Cf. A374189 (seen as set), A372728 (Kronecker), A373223 (Gauss), A374157, A004767.

Sequence in context: A135816 A157341 A038036 * A297793 A351177 A319880

Adjacent sequences: A374185 A374186 A374187 * A374189 A374190 A374191

Keyword

nonn,tabl

Author

Peter Luschny, Jun 30 2024

Status

approved