A374157 -id:A374157 - cp's OEIS frontend

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

Peter Luschny, Jun 30 2024

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.

			  [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, ...

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

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

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);

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])

All terms are even.

A374189 The values of A374188, seen as an ordered set.

Original entry on oeis.org

2, 10, 12, 18, 24, 26, 28, 34, 42, 44, 48, 50, 56, 58, 60, 66, 74, 76, 82, 88, 90, 92, 96, 98, 106, 108, 112, 114, 120, 122, 124, 130, 138, 140, 146, 152, 154, 156, 162, 170, 172, 176, 178, 184, 186, 188, 192, 194, 202, 204, 210, 216, 218, 220, 224, 226, 234, 236, 240

Offset: 1

Peter Luschny, Jul 01 2024

nonn

k is a term if and only if there exists a positive integer a of the form 4*n - 1, such that K(a / k) != K((-1)^floor(k/2)*k / a), where K denotes the Kronecker symbol (A372728).

			34 is a term, because K(15/34) = 1, but K(-34/15) = -1.
18 is a term, because K(19/18) = 1, but K(-18/19) = -1.

Cf. A374188, A372728, A374157.

All terms are even.

cp's OEIS Frontend

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.

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