A374184 -id:A374184 - 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.

A374180 Numbers k such that K(3 / k) != K((-1)^floor(k/2)*k / 3), where K(a/b) is the Kronecker symbol. Row 1 of A374188.

Original entry on oeis.org

2, 10, 26, 28, 34, 44, 50, 56, 58, 74, 76, 82, 88, 92, 98, 106, 112, 122, 124, 130, 140, 146, 152, 154, 170, 172, 176, 178, 184, 188, 194, 202, 218, 220, 224, 226, 236, 242, 248, 250, 266, 268, 274, 280, 284, 290, 298, 304, 314, 316, 322, 332, 338, 344, 346

Offset: 1

Peter Luschny, Jun 30 2024

nonn

Cf. A374188, A374181, A374182, A374183, A374184.

Cf. A372728 (Kronecker).

# see A374188
print(A374188_row(1, 350))

A374181 Numbers k such that K(7 / k) != K((-1)^floor(k/2)*k / 7), where K(a/b) is the Kronecker symbol. Row 2 of A374188.

Original entry on oeis.org

2, 10, 12, 18, 24, 26, 34, 44, 48, 50, 58, 60, 66, 74, 76, 82, 88, 90, 92, 96, 106, 108, 114, 120, 122, 124, 130, 138, 146, 152, 156, 162, 170, 172, 176, 178, 184, 186, 188, 192, 194, 202, 204, 216, 218, 220, 226, 234, 236, 240, 242, 248, 250, 258, 268, 274

Offset: 1

Peter Luschny, Jun 30 2024

nonn

Cf. A374188, A374180, A374182, A374183, A374184.

Cf. A372728 (Kronecker).

# see A374188
print(A374188_row(2, 350))

A374182 Numbers k such that K(11 / k) != K((-1)^floor(k/2)*k / 11), where K(a/b) is the Kronecker symbol. Row 3 of A374188.

Original entry on oeis.org

2, 10, 12, 18, 24, 26, 28, 34, 42, 48, 50, 56, 58, 60, 74, 76, 82, 90, 92, 96, 98, 106, 108, 112, 114, 120, 122, 124, 130, 138, 140, 146, 152, 156, 162, 170, 172, 178, 184, 186, 188, 192, 194, 202, 204, 210, 216, 218, 224, 226, 234, 236, 240, 248, 250, 252

Offset: 1

Peter Luschny, Jun 30 2024

nonn

Cf. A374188, A374180, A374181, A374183, A374184.

Cf. A372728 (Kronecker).

# see A374188
print(A374188_row(3, 350))

A374183 Numbers k such that K(15 / k) != K((-1)^floor(k/2)*k / 15), where K(a/b) is the Kronecker symbol. Row 4 of A374188.

Original entry on oeis.org

2, 26, 28, 34, 44, 56, 58, 74, 76, 82, 88, 92, 98, 106, 112, 122, 124, 146, 152, 154, 172, 176, 178, 184, 188, 194, 202, 218, 224, 226, 236, 242, 248, 266, 268, 274, 284, 298, 304, 314, 316, 322, 332, 338, 344, 346, 352, 362, 364, 368, 376, 386, 394, 412, 418

Offset: 1

Peter Luschny, Jun 30 2024

nonn

Cf. A374188, A374180, A374181, A374182, A374184.

Cf. A372728 (Kronecker).

# see A374188
print(A374188_row(4, 350))

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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A374180 Numbers k such that K(3 / k) != K((-1)^floor(k/2)*k / 3), where K(a/b) is the Kronecker symbol. Row 1 of A374188.

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A374181 Numbers k such that K(7 / k) != K((-1)^floor(k/2)*k / 7), where K(a/b) is the Kronecker symbol. Row 2 of A374188.

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SageMath

A374182 Numbers k such that K(11 / k) != K((-1)^floor(k/2)*k / 11), where K(a/b) is the Kronecker symbol. Row 3 of A374188.

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SageMath

A374183 Numbers k such that K(15 / k) != K((-1)^floor(k/2)*k / 15), where K(a/b) is the Kronecker symbol. Row 4 of A374188.

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Programs

SageMath