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.

A105885 Primes for which -13 is a primitive root.

Original entry on oeis.org

2, 3, 5, 23, 37, 41, 43, 73, 79, 89, 97, 107, 109, 127, 131, 137, 139, 149, 179, 191, 197, 199, 241, 251, 263, 281, 283, 293, 317, 347, 349, 353, 367, 397, 401, 439, 449, 457, 467, 491, 503, 509, 523, 541, 557, 563, 571, 577, 593, 599, 607, 613, 617, 647, 659, 719, 727
Offset: 1

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Author

N. J. A. Sloane, Apr 24 2005

Keywords

Comments

Primes p such that Legendre(-13,p) = -1. - N. J. A. Sloane, Dec 26 2017

Links

Programs

A296927 Inert rational primes in the field Q(sqrt(-13)).

Original entry on oeis.org

3, 5, 23, 37, 41, 43, 73, 79, 89, 97, 103, 107, 109, 127, 131, 137, 139, 149, 179, 191, 193, 197, 199, 211, 229, 241, 251, 263, 281, 283, 293, 311, 317, 347, 349, 353, 367, 397, 401, 409, 419, 421, 439, 443, 449, 457, 461, 467, 491, 503, 509, 523, 541, 547, 557, 563
Offset: 1

Views

Author

N. J. A. Sloane, Dec 26 2017

Keywords

Links

Programs

A296928 Primes p such that Legendre(-13,p) = 0 or 1.

Original entry on oeis.org

7, 11, 13, 17, 19, 29, 31, 47, 53, 59, 61, 67, 71, 83, 101, 113, 151, 157, 163, 167, 173, 181, 223, 227, 233, 239, 257, 269, 271, 277, 307, 313, 331, 337, 359, 373, 379, 383, 389, 431, 433, 463, 479, 487, 499, 521, 569, 587, 601, 619, 631
Offset: 1

Views

Author

N. J. A. Sloane, Dec 26 2017

Keywords

Comments

Primes == 1, 7, 9, 11, 13, 15, 17, 19, 25, 29, 31, 47, or 49 (mod 52). - Robert Israel, Dec 27 2017

Links

Programs

  • Maple
    Load the Maple program HH given in A296920. Then run HH(-13, 200); This produces A296926, A296927, A296928, A105885.
select(isprime, [seq(seq(52*i+j, j=[1, 7, 9, 11, 13, 15, 17, 19, 25, 29, 31, 47, 49]),i=0..50)]); # Robert Israel, Dec 27 2017

A035177 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -13.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A296926.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, KroneckerSymbol[-13, #] &]; Array[a, 100] (* Amiram Eldar, Nov 17 2023 *)
  • PARI
    my(m = -13); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(-13, d)); \\ Amiram Eldar, Nov 17 2023

Formula

From Amiram Eldar, Nov 17 2023 (Start)
a(n) = Sum_{d|n} Kronecker(-13, d).
Multiplicative with a(13^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(-13, p) = -1, and a(p^e) = e+1 if Kronecker(-13, p) = 1 (p is in A296926).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(13)) = 0.5808806... . (End)
Showing 1-4 of 4 results.

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

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