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.

A038882 Primes that are not in A038881.

Original entry on oeis.org

2, 3, 13, 17, 23, 29, 31, 41, 47, 59, 61, 67, 71, 73, 101, 103, 109, 149, 163, 173, 179, 191, 193, 197, 199, 223, 233, 241, 251, 277, 281, 293, 311, 331, 337, 349, 367, 373, 379, 383, 409, 419, 443, 457, 461, 463
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Also, only entries p == 1 (mod 4) of the sequence are not squares mod 11 (from the quadratic reciprocity law). - Lekraj Beedassy, Jul 21 2004
Except for 2, inert primes in Z[sqrt(11)]. 2 splits as (-1)*(3 - sqrt(11))*(3 + sqrt(11)). Cf. A296936. - Alonso del Arte, Jan 02 2015

Links

Crossrefs

Cf. A019339, A038881, A296936.

Programs

  • Mathematica
    Select[Prime@Range[120], JacobiSymbol[11, #] == -1 &] (* Vincenzo Librandi, Sep 08 2012 *)
  • PARI
    isok(p) = isprime(p) && !((p%2) && issquare(Mod(11, p))); \\ Michel Marcus, Jul 04 2023

Extensions

Offset changed from 0 to 1 by Vincenzo Librandi, Sep 08 2012
Definition edited by N. J. A. Sloane, Jul 04 2023

A035193 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 11.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A296935, A296936.

Programs

  • Mathematica
    a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[11, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)
  • PARI
    my(m=11); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(11, d)); \\ Amiram Eldar, Nov 18 2023

Formula

From Amiram Eldar, Nov 18 2023: (Start)
a(n) = Sum_{d|n} Kronecker(11, d).
Multiplicative with a(11^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(11, p) = -1 (p is in A296936), and a(p^e) = e+1 if Kronecker(11, p) = 1 (p is in A296935).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(3*sqrt(11)+10)/(3*sqrt(11)) = 0.60166042997... . (End)

A296935 Rational primes that decompose in the field Q(sqrt(11)).

Original entry on oeis.org

5, 7, 19, 37, 43, 53, 79, 83, 89, 97, 107, 113, 127, 131, 137, 139, 151, 157, 167, 181, 211, 227, 229, 239, 257, 263, 269, 271, 283, 307, 313, 317, 347, 353, 359, 389, 397, 401, 421, 431, 433, 439, 449, 479, 491, 503, 509, 521, 523, 547, 563, 571, 577, 607
Offset: 1

Views

Author

N. J. A. Sloane, Dec 26 2017

Keywords

Links

Crossrefs

Cf. A296936, A038881, A038882.

Programs

  • Maple
    Load the Maple program HH given in A296920. Then run HH(11, 200); This produces A296935, A296936, A038881, A038882.
  • Mathematica
    Select[Prime[Range[120]], KroneckerSymbol[11, #] == 1 &] (* Amiram Eldar, Dec 24 2023 *)
  • PARI
    list(lim)=my(v=List()); forprime(p=5,lim, if(kronecker(11,p)==1, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Mar 18 2018

A035226 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = 44.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A296935, A296936.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, KroneckerSymbol[44, #] &]; Array[a, 100] (* Amiram Eldar, Nov 20 2023 *)
  • PARI
    my(m = 44); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(44, d)); \\ Amiram Eldar, Nov 20 2023

Formula

From Amiram Eldar, Nov 20 2023: (Start)
a(n) = Sum_{d|n} Kronecker(44, d).
Multiplicative with a(p^e) = 1 if Kronecker(44, p) = 0 (p = 2 or 11), a(p^e) = (1+(-1)^e)/2 if Kronecker(44, p) = -1 (p is in A296936), and a(p^e) = e+1 if Kronecker(44, p) = 1 (p is in A296935).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(3*sqrt(11)+10)/sqrt(44) = 0.902490644956... . (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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