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 1-2 of 2 results.

A035219 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 37.

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 2, 0, 3, 0, 2, 2, 0, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 4, 2, 0, 0, 0, 0, 4, 0, 0, 3, 1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 2, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 4, 0, 0, 0, 5
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 37. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

Crossrefs

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A038914, A191027.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, KroneckerSymbol[37, #] &]; Array[a, 100] (* Amiram Eldar, Nov 20 2023 *)
  • PARI
    my(m = 37); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(37, d)); \\ Amiram Eldar, Nov 20 2023

Formula

From Amiram Eldar, Nov 20 2023: (Start)
a(n) = Sum_{d|n} Kronecker(37, d).
Multiplicative with a(37^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(37, p) = -1 (p is in A038914), and a(p^e) = e+1 if Kronecker(37, p) = 1 (p is in A191027).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(sqrt(37)+6)/sqrt(37) = 0.819292168725... . (End)

A035267 Indices of nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m= 37.

Original entry on oeis.org

1, 3, 4, 7, 9, 11, 12, 16, 21, 25, 27, 28, 33, 36, 37, 41, 44, 47, 48, 49, 53, 63, 64, 67, 71, 73, 75, 77, 81, 83, 84, 99, 100, 101, 107, 108, 111, 112, 121, 123, 127, 132, 137, 139, 141, 144, 147, 148, 149, 151, 157, 159, 164, 169, 173, 175, 176, 181, 188
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Positive numbers represented by the indefinite quadratic form 3x^2+xy-3y^2, of discriminant 37. - N. J. A. Sloane, Jun 05 2014 [Typo corrected by Klaus Purath, Apr 24 2023]
Also positive numbers of the form x^2 + (2m+1)xy + (m^2+m-9)y^2, m, x, y integers. All squares as well as the products of any terms belong to the sequence. Thus, this set of terms is closed under multiplication. - Klaus Purath, Apr 24 2023
A positive integer k belongs to the sequence if and only if k (modulo 37) is a term of A010398 and, moreover, in the case that prime factors p of k are terms of A038914, they occur only with even exponents. Or, more briefly, any positive integer is a term of this sequence if none of its divisors is an odd power of primes from A038914. For these primes also p (modulo 37) = {2, 5, 6, 8, 13, ...} = A028750 applies. - Klaus Purath, May 12 2023

Links

Crossrefs

For primes see A141178.
Cf. A035219.

Programs

  • Mathematica
    Reap[For[n = 0, n <= 100, n++, If[ Reduce[ 3*x^2 + x*y - 3*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]] (* N. J. A. Sloane, Jun 05 2014 *)
  • PARI
    m=37; select(x -> x, direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020

Extensions

More terms from Colin Barker, Jun 17 2014
Showing 1-2 of 2 results.

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

Content is available under The OEIS End-User License Agreement.