A374189 -id:A374189 - cp's OEIS frontend

A374190 Complement of A374189 in A005843.

0, 4, 6, 8, 14, 16, 20, 22, 30, 32, 36, 38, 40, 46, 52, 54, 62, 64, 68, 70, 72, 78, 80, 84, 86, 94, 100, 102, 104, 110, 116, 118, 126, 128, 132, 134, 136, 142, 144, 148, 150, 158, 160, 164, 166, 168, 174, 180, 182, 190, 196, 198, 200, 206, 208, 212, 214, 222

Offset: 1

Peter Luschny, Jul 02 2024

nonn

k is a term if and only if for all a of the form 4*n - 1 (n>=1) there is an even k such that K(a / k) = K((-1)^floor(k/2)*k / a), where K denotes the Kronecker symbol (A372728).

Cf. A374189, A005843 (even numbers).

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.

Original entry on oeis.org

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.

A374187 Least a of the form 4n - 1 (n>=1) such that there is a positive integer k so that K(a / k) != K((-1)^floor(k/2)k / a), where K denotes the Kronecker symbol (A372728).

Original entry on oeis.org

3, 3, 7, 7, 7, 3, 3, 3, 11, 3, 7, 3, 3, 3, 7, 7, 3, 3, 3, 3, 7, 3, 7, 3, 3, 7, 3, 7, 7, 3, 3, 3, 7, 3, 3, 3, 3, 7, 7, 3, 3, 3, 3, 3, 7, 3, 7, 3, 3, 7, 11, 7, 3, 3, 3, 3, 7, 3, 7, 3, 3, 3, 11, 7, 3, 3, 3, 3, 7, 3, 3, 3, 7, 3, 7, 7, 3, 3, 3, 7, 3, 3, 3, 3, 7, 3

Offset: 1

Peter Luschny, Jul 02 2024

nonn

These values witness the correctness of A374189. They seem to grow very slowly.

Cf. A374189, A374188, A372728.

cp's OEIS Frontend

A374190 Complement of A374189 in A005843.

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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.

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SageMath

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A374187 Least a of the form 4n - 1 (n>=1) such that there is a positive integer k so that K(a / k) != K((-1)^floor(k/2)k / a), where K denotes the Kronecker symbol (A372728).

Views

Author

Keywords

Comments

Crossrefs

cp's OEIS Frontend

A374190 Complement of A374189 in A005843.

Views

Author

Keywords

Comments

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

SageMath

Formula

A374187 Least a of the form 4*n - 1 (n>=1) such that there is a positive integer k so that K(a / k) != K((-1)^floor(k/2)*k / a), where K denotes the Kronecker symbol (A372728).

Views

Author

Keywords

Comments

Crossrefs

A374187 Least a of the form 4n - 1 (n>=1) such that there is a positive integer k so that K(a / k) != K((-1)^floor(k/2)k / a), where K denotes the Kronecker symbol (A372728).