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.

A296929 Rational primes that decompose in the field Q(sqrt(-17)).

Original entry on oeis.org

3, 7, 11, 13, 23, 31, 53, 71, 79, 89, 101, 107, 131, 137, 139, 149, 157, 163, 167, 199, 211, 227, 229, 257, 281, 283, 293, 311, 347, 349, 353, 367, 373, 379, 389, 409, 419, 421, 431, 433, 439, 457, 461, 479, 487, 499, 503, 509, 547, 557, 569, 571, 577, 593, 607
Offset: 1

Views

Author

N. J. A. Sloane, Dec 26 2017

Keywords

Links

Programs

Formula

Primes == {1, 3, 7, 9, 11, 13, 21, 23, 25, 27, 31, 33, 39, 49, 53, 63} (mod 68). - Travis Scott, Jan 05 2023

A296931 Primes p such that Legendre(-17,p) = 0 or 1.

Original entry on oeis.org

2, 3, 7, 11, 13, 17, 23, 31, 53, 71, 79, 89, 101, 107, 131, 137, 139, 149, 157, 163, 167, 199, 211, 227, 229, 257, 281, 283, 293, 311, 347, 349, 353, 367, 373, 379, 389, 409, 419, 421, 431, 433, 439, 457, 461, 479, 487, 499, 503, 509, 547, 557
Offset: 1

Views

Author

N. J. A. Sloane, Dec 26 2017

Keywords

Comments

Primes p such that p == 1, 2, 3, 7, 9, 11, 13, 17, 21, 23, 25, 27, 31, 33, 39, 49, 53, or 63 (mod 68). - Robert Israel, Dec 26 2017

Links

Programs

  • Maple
    Load the Maple program HH given in A296920. Then run HH(-17, 200); This produces A296929, A296930, A296931.
Alternative:
select(p-> isprime(p) and numtheory:-legendre(-17,p)<>-1, [2,seq(i,i=3..1000)]); # Robert Israel, Dec 26 2017

A035173 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -17.

Original entry on oeis.org

1, 2, 2, 3, 0, 4, 2, 4, 3, 0, 2, 6, 2, 4, 0, 5, 1, 6, 0, 0, 4, 4, 2, 8, 1, 4, 4, 6, 0, 0, 2, 6, 4, 2, 0, 9, 0, 0, 4, 0, 0, 8, 0, 6, 0, 4, 0, 10, 3, 2, 2, 6, 2, 8, 0, 8, 0, 0, 0, 0, 0, 4, 6, 7, 0, 8, 0, 3, 4, 0, 2, 12, 0, 0, 2, 0, 4, 8, 2, 0, 5
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A296930.

Programs

  • Mathematica
    a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-17, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Jul 17 2018 *)
  • PARI
    my(m = -17); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(-17, d)); \\ Amiram Eldar, Nov 17 2023

Formula

From Amiram Eldar, Nov 17 2023: (Start)
a(n) = Sum_{d|n} Kronecker(-17, d).
Multiplicative with a(17^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(-17, p) = -1 (p is in A296930), and a(p^e) = e+1 if Kronecker(-17, p) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/sqrt(17) = 3.047792... . (End)

A369863 Inert rational primes in the field Q(sqrt(-21)).

Original entry on oeis.org

13, 29, 43, 47, 53, 59, 61, 67, 73, 79, 83, 97, 113, 127, 131, 137, 149, 151, 157, 163, 167, 181, 197, 211, 227, 229, 233, 241, 251, 281, 311, 313, 317, 331, 349, 379, 383, 389, 397, 401, 409, 419, 433, 449, 463, 467, 479, 487, 499, 503, 547, 557, 563, 569, 571, 577, 587
Offset: 1

Views

Author

Dimitris Cardaris, Feb 03 2024

Keywords

Comments

Primes p such that Legendre(-21,p) = -1.

Crossrefs

Cf. inert rational primes in the imaginary quadratic field Q(sqrt(-d)) for the first squarefree positive integers d: A002145 (1), A003628 (2), A003627 (3), A003626 (5), A191059 (6), A003625 (7), A296925 (10), A191060 (11), A105885 (13), A191061 (14), A191062 (15), A296930 (17), A191063 (19), this sequence (21), A191064 (22), A191065 (23).

Programs

  • Mathematica
    Select[Range[3,600], PrimeQ[#] && JacobiSymbol[-21,#]==-1 &] (* Stefano Spezia, Feb 04 2024 *)
  • SageMath
    [p for p in prime_range(3, 600) if legendre_symbol(-21, p) == -1]
Showing 1-4 of 4 results.

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

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