cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A322829 a(n) = Jacobi (or Kronecker) symbol (n/21).

Original entry on oeis.org

0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0
Offset: 0

Views

Author

Jianing Song, Dec 27 2018

Keywords

Comments

Period 21: repeat [0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1].
Also a(n) = Kronecker symbol (21/n).
This sequence is one of the three non-principal real Dirichlet characters modulo 21. The other two are Jacobi or Kronecker symbols {(n/63)} (or {(-63/n)}) and {(n/147)} (or {(-147/n)}).

Links

Crossrefs

Cf. A035203 (inverse Moebius transform).
Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017(d=-24), A011586 (d=-23), A289741 (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), A101455 (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), this sequence (d=21), A322796 (d=24).

Programs

  • Mathematica
    JacobiSymbol[Range[0, 100], 21] (* Paolo Xausa, Mar 19 2025 *)
  • PARI
    a(n) = kronecker(n, 21)

Formula

a(n) = 1 for n == 1, 4, 5, 16, 17, 20 (mod 21); -1 for n == 2, 8, 10, 11, 13, 19 (mod 21); 0 for n that are not coprime with 21.
Completely multiplicative with a(p) = a(p mod 21) for primes p.
a(n) = A102283(n)*A175629(n).
a(n) = a(n+21) = -a(n) for all n in Z.
From Chai Wah Wu, Feb 18 2021: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-6) + a(n-8) - a(n-9) + a(n-11) - a(n-12) for n > 11.
G.f.: -x*(x - 1)*(x + 1)*(x^8 - 2*x^7 + 2*x^6 + 2*x^2 - 2*x + 1)/(x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1). (End)

A327655 Intersection of A327653 and A327654.

Original entry on oeis.org

119, 649, 1189, 4187, 12871, 14041, 16109, 23479, 24769, 28421, 31631, 34997, 38503, 41441, 48577, 50545, 56279, 58081, 59081, 61447, 75077, 91187, 95761, 96139, 116821, 127937, 146329, 148943, 150281, 157693, 170039, 180517, 188501, 207761, 208349, 244649, 281017, 311579, 316409
Offset: 1

Views

Author

Jianing Song, Sep 20 2019

Keywords

Comments

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n) = m*x(n-1) + x(n-2) for k >= 2. For primes p, we have (a) p divides x(p-((m^2+4)/p)); (b) x(p) == ((m^2+4)/p) (mod p), where (D/p) is the Kronecker symbol. This sequence gives composite numbers k such that gcd(k, m^2+4) = 1 and that conditions similar to (a) and (b) hold for k simultaneously, where m = 2.
If k is not required to be coprime to m^2 + 4 (= 13), then there are 322 such k <= 10^5 and 1381 such k <= 10^6, while there are only 24 terms <= 10^5 and 72 terms <= 10^6 in this sequence.

Examples

			119 divides A006190(120) as well as A006190(119) + 1, so 119 is a term.

Crossrefs

m m=1 m=2 m=3
k | x(k-Kronecker(m^2+4,k))* A081264 U A141137 A327651 A327653
k | x(k)-Kronecker(m^2+4,k) A049062 A099011 A327654
both A212424 A327652 this seq
* k is composite and coprime to m^2 + 4.
Cf. A006190, A011583 ({Kronecker(13,n)}).

Programs

  • PARI
    seqmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
isA327655(n)=!isprime(n) && seqmod(n, n)==kronecker(13,n) && !seqmod(n-kronecker(13,n), n) && gcd(n,13)==1 && n>1

A140782 a(n) = sigma(n) * Kronecker(13, n).

Original entry on oeis.org

1, -3, 4, 7, -6, -12, -8, -15, 13, 18, -12, 28, 0, 24, -24, 31, 18, -39, -20, -42, -32, 36, 24, -60, 31, 0, 40, -56, 30, 72, -32, -63, -48, -54, 48, 91, -38, 60, 0, 90, -42, 96, 44, -84, -78, -72, -48, 124, 57, -93, 72, 0, 54, -120, 72, 120, -80, -90, -60, -168, 62, 96, -104, 127, 0, 144, -68, 126, 96, -144
Offset: 1

Views

Author

Michael Somos, Jun 04 2008

Keywords

Comments

In the notation of Parry 1979 page 166, the g.f. is (theta_1 - theta_2) / 2 + theta_3 - theta_4 + theta_5 - theta_6 + theta_7 - theta_8 where theta_k is g.f. for A107497, ..., A107504.

Examples

			q - 3*q^2 + 4*q^3 + 7*q^4 - 6*q^5 - 12*q^6 - 8*q^7 - 15*q^8 + 13*q^9 + ...

Links

Crossrefs

Cf. A000203, A011583, A107497, A107498, A107499, A107500, A107501, A107502, A107503, A107504.

Programs

  • Mathematica
    Table[If[n==0, 0, DivisorSigma[1, n] JacobiSymbol[13, n]], {n, 100}] (* Indranil Ghosh, Jul 02 2017 *)
  • PARI
    {a(n) = if( n==0, 0, sigma(n) * kronecker( 13, n))}
  • PARI
    {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; (p^(e+1) - 1) / (p - 1) * kronecker( 13, p)^e)))}

Formula

a(n) is multiplicative with a(p^e) = (p^(e+1) - 1) / (p - 1) * Kronecker(13, p)^e.
G.f. is a period 1 Fourier series which satisfies f(-1 / (169 t)) = -169 (t/i)^2 f(t) where q = exp(2 Pi i t).
a(13*n) = 0. a(n) = A000203(n) * A011583(n). |a(n)| = A000203(n) unless 13 divides n.
a(n) = (A107497(n) - A107498(n)) / 2 + A107499(n) - A107500(n) + A107501(n) - A107502(n) + A107503(n) - A107504(n).
Previous Showing 11-13 of 13 results.

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