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.

Showing 1-4 of 4 results.

A035184 a(n) = Sum_{d|n} Kronecker(-1, d).

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 0, 4, 1, 4, 0, 0, 2, 0, 0, 5, 2, 2, 0, 6, 0, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 6, 0, 4, 0, 3, 2, 0, 0, 8, 2, 0, 0, 0, 2, 0, 0, 0, 1, 6, 0, 6, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 7, 4, 0, 0, 6, 0, 0, 0, 4, 2, 4, 0, 0, 0, 0, 0, 10, 1, 4, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 2, 2, 0, 9, 2, 0, 0, 8, 0
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			G.f. = x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^8 + x^9 + 4*x^10 + 2*x^13 + 5*x^16 + 2*x^17 + ...

Links

Crossrefs

Cf. A002175, A008441, A019669, A053692, A113407, A121444.
Inverse Moebius transform of A034947.
Sum_{d|n} Kronecker(k, d): A035143..A035181 (k=-47..-9, skipping numbers that are not cubefree), A035182 (k=-7), A192013 (k=-6), A035183 (k=-5), A002654 (k=-4), A002324 (k=-3), A002325 (k=-2), this sequence (k=-1), A000012 (k=0), A000005 (k=1), A035185 (k=2), A035186 (k=3), A001227 (k=4), A035187..A035229 (k=5..47, skipping numbers that are not cubefree).

Programs

  • Mathematica
    a[n_] := DivisorSum[n, KroneckerSymbol[-1, #] &]; Array[a, 105] (* Jean-François Alcover, Dec 02 2015 *)
  • PARI
    {a(n) = if( n<1, 0, direuler( p=2, n, 1/((1 - X) * (1 - kronecker( -1, p) * X))) [n])}; /* Michael Somos, Jan 05 2012 */
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -1, d)))}; /* Michael Somos, Jan 05 2012 */

Formula

a(n) is multiplicative with a(2^e) = e + 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4). - Michael Somos, Jan 05 2012
a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). a(12*n + 1) = A002175(n). a(12*n + 5) = 2 * A121444(n).
Dirichlet g.f.: zeta(s)*beta(s)/(1 - 2^(-s)), where beta is the Dirichlet beta function. - Ralf Stephan, Mar 27 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 = 1.570796... (A019669). - Amiram Eldar, Oct 17 2022

A035181 a(n) = Sum_{d|n} Kronecker(-9, d).

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 0, 4, 1, 4, 0, 3, 2, 0, 2, 5, 2, 2, 0, 6, 0, 0, 0, 4, 3, 4, 1, 0, 2, 4, 0, 6, 0, 4, 0, 3, 2, 0, 2, 8, 2, 0, 0, 0, 2, 0, 0, 5, 1, 6, 2, 6, 2, 2, 0, 0, 0, 4, 0, 6, 2, 0, 0, 7, 4, 0, 0, 6, 0, 0, 0, 4, 2, 4, 3, 0, 0, 4, 0, 10, 1, 4, 0, 0, 4, 0, 2, 0, 2, 4, 0, 0, 0, 0, 0, 6, 2, 2, 0, 9, 2, 4, 0, 8, 0
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			x + 2*x^2 + x^3 + 3*x^4 + 2*x^5 + 2*x^6 + 4*x^8 + x^9 + 4*x^10 + 3*x^12 + ...

Links

Crossrefs

Cf. A008441, A019693, A125079.
Sum_{d|n} Kronecker(k, d): A035143..A035181 (k=-47..-9, skipping numbers that are not cubefree), A035182 (k=-7), A192013 (k=-6), A035183 (k=-5), A002654 (k=-4), A002324 (k=-3), A002325 (k=-2), A035184 (k=-1), A000012 (k=0), A000005 (k=1), A035185 (k=2), A035186 (k=3), A001227 (k=4), A035187..A035229 (k=5..47, skipping numbers that are not cubefree).

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -9, d], { d, Divisors[ n]}]] (* Michael Somos, Jun 24 2011 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -9, d)))} \\ Michael Somos, Jun 24 2011
  • PARI
    {a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -9, p) * X))) [n])} \\ Michael Somos, Jun 24 2011
  • PARI
    {a(n) = local(A, p, e); if( n<0, 0, A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if( p==2, e+1, if( p==3, 1, if( p%4==1, e+1, (1 + (-1)^e)/2))))))} \\ Michael Somos, Jun 24 2011
  • PARI
    A035181(n)=sumdivmult(n,d,kronecker(-9,d)) \\ M. F. Hasler, May 08 2018

Formula

From Michael Somos, Jun 24 2011: (Start)
a(n) is multiplicative with a(2^e) = e + 1, a(3^e) = 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4) and p > 3.
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker(-9, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker(-9, p) * p^-s)). (End)
a(3*n) = a(n). a(2*n + 1) = A125079(n). a(4*n + 1) = A008441(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/3 = 2.094395... (A019693). - Amiram Eldar, Oct 17 2022

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

Views

Author

Wolfdieter Lang, May 29 2013

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Maple
    0, seq(numtheory:-jacobi(-5, n), n=1..89); # Peter Luschny, Dec 30 2018
  • Mathematica
    Table[KroneckerSymbol[-5, n],{n,0,89}]
  • PARI
    a(n)=kronecker(-5,n); \\ Andrew Howroyd, Jul 23 2018

Formula

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

Extensions

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A035249 Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m= -5.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 12, 15, 16, 20, 21, 23, 25, 27, 28, 29, 35, 36, 41, 43, 45, 47, 48, 49, 60, 61, 63, 64, 67, 69, 75, 80, 81, 83, 84, 87, 89, 92, 100, 101, 103, 105, 107, 108, 109, 112, 115, 116, 121, 123, 125, 127, 129, 135, 140, 141, 144, 145, 147, 149, 161
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Crossrefs

Cf. A035183 (the expansion itself).

Programs

  • PARI
    m=-5; select(x -> x, direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X)), 1)

Extensions

Edited and extended by Andrey Zabolotskiy, Jul 30 2020
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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