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 1-5 of 5 results.

A377175 Primes p such that -5/2 is a primitive root modulo p.

Original entry on oeis.org

3, 17, 31, 43, 67, 71, 73, 79, 83, 101, 107, 109, 113, 137, 149, 163, 191, 199, 227, 229, 233, 239, 257, 269, 271, 283, 307, 311, 313, 337, 347, 349, 353, 359, 389, 421, 431, 433, 439, 443, 461, 467, 479, 509, 547, 563, 587, 593, 599, 617, 631, 661, 673, 683, 719, 821, 827, 829, 839, 857, 907, 911, 919, 941, 947, 953, 977
Offset: 1

Views

Author

Jianing Song, Oct 18 2024

Keywords

Comments

If p is a term in this sequence, then -5/2 is not a square modulo p (i.e., p is in A296925).
Conjecture: this sequence has relative density equal to Artin's constant (A005596) with respect to the set of primes.

Links

Crossrefs

Primes p such that +a/2 is a primitive root modulo p: A320384 (a=3), A377174 (a=5), A377176 (a=7), A377178 (a=9).
Primes p such that -a/2 is a primitive root modulo p: A377172 (a=3), this sequence (a=5), A377177 (a=7), A377179 (a=9).
Cf. A296925, A005596.

Programs

  • PARI
    print1(3, ", "); forprime(p=7, 10^3, if(znorder(Mod(-5/2, p))==p-1, print1(p, ", ")));

A155488 Primes p with property that p^2 is of the form x^2 + 40y^2.

Original entry on oeis.org

7, 11, 13, 19, 23, 37, 41, 47, 53, 59, 89, 103, 127, 131, 139, 157, 167, 173, 179, 197, 211, 223, 241, 251, 263, 277, 281, 293, 317, 331, 367, 373, 379, 383, 397, 401, 409, 419, 449, 463, 487, 491, 499, 503, 521, 557, 569, 571, 601, 607, 613, 619, 641, 647
Offset: 1

Views

Author

Zak Seidov, Jan 23 2009

Keywords

Comments

All p^2 are congruent to {1, 9} (mod 40), as in A107145.
Rational primes that decompose in the field Q(sqrt(-10)). - N. J. A. Sloane, Dec 26 2017

Links

Crossrefs

Cf. A107145 (Primes of the form x^2 + 40y^2).

Programs

A293859 Prime factors of numbers of the form k^2 + 10.

Original entry on oeis.org

2, 5, 7, 11, 13, 19, 23, 37, 41, 47, 53, 59, 89, 103, 127, 131, 139, 157, 167, 173, 179, 197, 211, 223, 241, 251, 263, 277, 281, 293, 317, 331, 367, 373, 379, 383, 397, 401, 409, 419, 449, 463, 487, 491, 499, 503, 521, 557, 569, 571, 601, 607, 613, 619, 641
Offset: 1

Views

Author

J. Lowell, Oct 17 2017

Keywords

Comments

Primes p such that Legendre(-10,p) = 0 or 1. - N. J. A. Sloane, Dec 26 2017
Question: Is there a comment of the form "a prime number is in this sequence if and only if it is congruent to (list of appropriate values) mod n" for this sequence?
From Robert Israel, Nov 19 2017: (Start)
Prime p > 5 is in the sequence iff -10 is a quadratic residue mod p.
Thus p is either in the intersection of A002144 and A038879 or in neither of them.
Primes == 1, 2, 5, 7, 9, 11, 13, 19, 23, or 37 (mod 40). (End)

Examples

			7 is in the sequence because 2^2 + 10 = 14 is 2 times 7.
19 is in the sequence because 3^2 + 10 = 19.

Links

Crossrefs

Cf. A002144, A038879, A114948.

Programs

  • Maple
    select(isprime, [seq(seq(i*40+j, j = [1, 2, 5, 7, 9, 11, 13, 19, 23, 37]), i=0..40)]); # Robert Israel, Nov 19 2017
# Load the Maple program HH given in A296920. Then run HH(-10, 200); This produces A155488, A296925, A293859. - N. J. A. Sloane, Dec 26 2017
  • Mathematica
    Select[Prime@ Range@ 120, {} != FindInstance[# x == n^2 + 10 && n >= 0 && x > 0, {n, x}, Integers, 1] &] (* Giovanni Resta, Oct 19 2017 *)

Extensions

More terms from Giovanni Resta, Oct 19 2017

A035180 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -10.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A296925.

Programs

  • Mathematica
    a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-10, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)
  • PARI
    my(m=-10); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(-10, d)); \\ Amiram Eldar, Nov 17 2023

Formula

From Amiram Eldar, Nov 17 2023: (Start)
a(n) = Sum_{d|n} Kronecker(-10, d).
Multiplicative with a(p^e) = 1 if Kronecker(-10, p) = 0 (p = 2 or 5), a(p^e) = (1+(-1)^e)/2 if Kronecker(-10, p) = -1 (p is in A296925), and a(p^e) = e+1 if Kronecker(-10, p) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(10) = 0.993458... . (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 1-5 of 5 results.

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

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