A089509 -id:A089509 - cp's OEIS frontend

A074230 Numbers n such that A089509(n)=1.

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 19, 24, 25, 27, 29, 31, 32, 36, 37, 38, 47, 48, 50, 53, 54, 55, 57, 58, 59, 62, 64, 65, 72, 74, 75, 76, 81, 83, 85, 87, 93, 94, 96, 100, 103, 106, 108, 109, 110, 111, 113, 114, 115, 116, 118, 121, 124, 128, 130, 131, 137, 139

Offset: 1

Jon Perry, Sep 17 2002

nonn

The original name was: Kronecker(7,y) == mu(gcd(7,y)).

Matthew House, Table of n, a(n) for n = 1..10000

Cf. A089509.

select(t -> numtheory:-jacobi(7,t)=1, [$1..1000]); # Robert Israel, Jul 24 2016

for (x=1,200, for (y=1,200,if (kronecker(x,y) == moebius(gcd(x,y)),write("km.txt",x,";",y," : ",kronecker(x,y)))))

Name simplified by Robert Israel, Jul 24 2016.

A175629 Legendre symbol (n,7).

Original entry on oeis.org

0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1

Offset: 0

R. J. Mathar, Jul 29 2010

This represents a non-principal Dirichlet character modulo 7.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=7, Chi_2(n).

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,-1,-1).

Cf. A089509, A097343.
The Legendre symbols (n,p): A091337 (p = 2, Kronecker symbol), A102283 (p = 3), A080891 (p = 5), this sequence (p = 7), A011582 (p = 11), A011583 (p = 13), ..., A011631 (p = 251), A165573 (p = 257), A165574 (p = 263). Also, many other sequences for p > 263 are in the OEIS.
Moebius transform of A035182.

&cat [[0, 1, 1, -1, 1, -1, -1]^^20]; // Vincenzo Librandi, Jun 30 2018

A := proc(n) numtheory[jacobi](n,7) ; end proc: seq(A(n),n=0..120) ;

LinearRecurrence[{-1,-1,-1,-1,-1,-1},{0,1,1,-1,1,-1},100] (* or *) PadRight[ {},100,{0,1,1,-1,1,-1,-1}] (* Harvey P. Dale, Aug 02 2013 *)
Table[JacobiSymbol[n, 7], {n, 0, 100}] (* Vincenzo Librandi, Jun 30 2018 *)

a(n) = kronecker(n, 7); \\ Michel Marcus, Jan 28 2019

a(n) = a(n+7).
|a(n)| = A109720(n).
a(n) = -a(n-1) - a(n-2) - a(n-3) - a(n-4) - a(n-5) - a(n-6).
G.f.: x*(1 + 2*x + x^2 + 2*x^3 + x^4)/(1 + x + x^2 + x^3 + x^4 + x^5 + x^6).
a(n) == n^3 (mod 7). - Jianing Song, Jun 29 2018

A215200 Triangle read by rows, Kronecker symbol (n-k|k) for n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, -1, -1, 1, 0, 1, 0, 0, 0, 1, 0, 1, -1, 1, 1, -1, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0

Offset: 1

Peter Luschny, Aug 05 2012

Signed version of A054521.

			Triangle begins:
  1,
  1,  0,
  1,  1,  0,
  1,  0,  1, 0,
  1, -1, -1, 1,  0,
  1,  0,  0, 0,  1, 0,
  1, -1,  1, 1, -1, 1,  0,
  1,  0, -1, 0, -1, 0,  1, 0,
  1,  1,  0, 1,  1, 0,  1, 1, 0,
  1,  0,  1, 0,  0, 0, -1, 0, 1, 0,
From _Jianing Song_, Dec 26 2018: (Start)
This sequence can also be arranged into a square array T(n,k) = Kronecker symbol(n|k) with n >= 0, k >= 1, read by antidiagonals:
  1  0  0  0  0  0  0 ... ((0|k) = A000007(k+1))
  1  1  1  1  1  1  1 ... ((1|k) = A000012)
  1  0 -1  0 -1  0 -1 ... ((2|k) = A091337)
  1 -1  0  1 -1  0 -1 ... ((3|k) = A091338)
  1  0  1  0  1  0  1 ... ((4|k) = A000035)
  1 -1 -1  1  0  1 -1 ... ((5|k) = A080891)
  1  0  0  0  1  0 -1 ... ((6|k) = A322796)
  1  1  1  1 -1  1  0 ... ((7|k) = A089509)
  ... (End)

Henri Cohen: A Course in Computational Algebraic Number Theory, p. 29.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Eric Weisstein's World of Mathematics, Kronecker Symbol.

Rows of square array include: A000012, A091337, A091338, A000035, A080891, A322796, A089509.

Cf. A054521, A215283, A215284, A215285.

/* As triangle */ [[KroneckerSymbol(n-k, k):  k in [1..n]]: n in [1..21]]; // Vincenzo Librandi, Apr 24 2018

A215200_row := n -> seq(numtheory[jacobi](n-k,k),k=1..n);
for n from 1 to 13 do A215200_row(n) od;

Column[Table[KroneckerSymbol[n - k, k], {n, 10}, {k, n}], Center] (* Alonso del Arte, Aug 06 2012 *)

T(n,k) = kronecker(n-k, k);
tabl(nn) = for(n=1, nn, for(k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Apr 24 2018

def A215200_row(n): return [kronecker_symbol(n-k,k) for k in (1..n)]
for n in (1..13): print(A215200_row(n))

A226162 a(n) = Kronecker Symbol (-5/n), n >= 0.

Original entry on oeis.org

0, 1, -1, 1, 1, 0, -1, 1, -1, 1, 0, -1, 1, -1, -1, 0, 1, -1, -1, -1, 0, 1, 1, 1, -1, 0, 1, 1, 1, 1, 0, -1, -1, -1, 1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, -1, 1, 1, 1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 1, 1, 1, 1, 0, 1, 1, -1, 1, 0, -1, -1, -1, 1, 0, -1, -1, 1, -1, 0, 1, -1, 1, 1, 0, -1, 1, 1, 1

Offset: 0

Wolfdieter Lang, May 29 2013

The number of -1's among the four terms following the 0 at a(5*k), for k >= 0, is 1, 2, 3, 3, 1, 0, 3, 2, 2, 1, 4, 4, 0, 1, 3, 3, 1, 1, 3, 4, ...
See the Weisstein link, where it is stated that the period length is 0.
In general, the sequence {(k/n)} is not periodic if and only if k == 3 (mod 4). - Jianing Song, Dec 30 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, Leo Goldmakher, Mock characters and the Kronecker symbol, arXiv:1608.03957 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Kronecker Symbol (contains this sequence).
Wikipedia, Kronecker Symbol

Cf. A035183 (inverse Moebius transform).
Sequences related to Kronecker symbols that do not form a Dirichlet character: this sequence {(-5/n)}, A034947 {(-1/n)}, A091338 {(3/n)}, A089509 {(7/n)}.
Cf. A080891 (5/n), A100047.

0, seq(numtheory:-jacobi(-5, n), n=1..89); # Peter Luschny, Dec 30 2018

Mathematica
```
Table[KroneckerSymbol[-5, n],{n,0,89}]
```

a(n)=kronecker(-5,n); \\ Andrew Howroyd, Jul 23 2018

Completely multiplicative with a(2) = -1, a(5) = 0, a(p) = 1 if p == 1, 3, 7, 9 (mod 20), a(p) = -1 if p == 11, 13, 17, 19 (mod 20). - Jianing Song, Dec 30 2018

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A091399 a(n) = Product_{ p | n } (1 + Legendre(7,p) ).

Original entry on oeis.org

1, 2, 2, 2, 0, 4, 1, 2, 2, 0, 0, 4, 0, 2, 0, 2, 0, 4, 2, 0, 2, 0, 0, 4, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 4, 2, 4, 0, 0, 0, 4, 0, 0, 0, 0, 2, 4, 1, 0, 0, 0, 2, 4, 0, 2, 4, 4, 2, 0, 0, 4, 2, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 4, 0, 0, 0, 0, 2, 0, 2, 4, 0, 0, 4, 0, 0, 0, 0, 0, 4, 4, 0, 4, 0, 2, 0, 0, 0, 0, 2, 0, 0

Offset: 1

N. J. A. Sloane, Mar 02 2004

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A089509, A091379, A091396.

with(numtheory); L := proc(n,N) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul((1+legendre(N,t1[i][1])),i=1..nops(t1)); end; [seq(L(n,7),n=1..120)];

a[1] = 1; a[n_] := Product[1+JacobiSymbol[7, p], {p, FactorInteger[n][[All, 1]]}];
Array[a, 105] (* Jean-François Alcover, Aug 26 2019 *)

a(n)={my(f=factor(n)[,1]); prod(i=1, #f, 1 + kronecker(7, f[i]))} \\ Andrew Howroyd, Jul 23 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 7*sqrt(7) * log(8+3*sqrt(7))/(4*Pi^2) = 1.298843... . - Amiram Eldar, Oct 17 2022

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A074230 Numbers n such that A089509(n)=1.

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A175629 Legendre symbol (n,7).

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A215200 Triangle read by rows, Kronecker symbol (n-k|k) for n >= 1, 1 <= k <= n.

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A226162 a(n) = Kronecker Symbol (-5/n), n >= 0.

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A091399 a(n) = Product_{ p | n } (1 + Legendre(7,p) ).

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