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 1-3 of 3 results.

A097955 Primes p such that p divides 5^((p-1)/2) - 2^((p-1)/2).

Original entry on oeis.org

3, 13, 31, 37, 41, 43, 53, 67, 71, 79, 83, 89, 107, 151, 157, 163, 173, 191, 197, 199, 227, 239, 241, 271, 277, 281, 283, 293, 307, 311, 317, 347, 359, 373, 397, 401, 409, 431, 439, 443, 449, 467, 479, 521, 523, 547, 557, 563, 569, 587, 599, 601, 613, 631, 641
Offset: 1

Views

Author

Cino Hilliard, Sep 06 2004

Keywords

Comments

Also 3 and primes p such that (p^2 - 1)/24 mod 10 = {0, 7}. - Richard R. Forberg, Aug 31 2013
Also primes p such that x^2 = 10 mod p has integer solutions, or Legendre(10, p) = 1. However, p could be irreducible but not prime in Z[sqrt(10)], especially if p = 3 or 7 mod 10. - Alonso del Arte, Dec 27 2015
Rational primes that decompose in the field Q(sqrt(10)). - N. J. A. Sloane, Dec 26 2017
From Jianing Song, Oct 13 2022: (Start)
Primes p such that kronecker(10,p) = 1 (or equivalently, kronecker(40,p) = 1).
Primes congruent to 1, 3, 9, 13, 27, 31, 37, 39 modulo 40. (End)

Examples

			For p = 13, 5^6 - 2^6 = 15561 is divisible by 13, so 13 is in the sequence.

Links

Crossrefs

A038879, the sequence of primes that do not remain inert in the field Q(sqrt(10)), is essentially the same.
Cf. A038880 (rational primes that remain inert in the field Q(sqrt(10))).

Programs

  • Maple
    select(p -> isprime(p) and  10 &^ ((p-1)/2)  mod p = 1, [seq(i,i=3..1000,2)]); # Robert Israel, Dec 28 2015
  • Mathematica
    Select[Prime[Range[100]], JacobiSymbol[10, #] == 1 &] (* Alonso del Arte, Dec 27 2015 *)
  • PARI
    \\ s = +-1,d=diff
ptopm1d2(n,x,d,s) = { forprime(p=3,n,p2=(p-1)/2; y=x^p2 + s*(x-d)^p2; if(y%p==0,print1(p","))) }
ptopm1d2(1000, 5, 3, -1)
  • PARI
    isA097955(p) == isprime(p) && kronecker(10,p) == 1 \\ Jianing Song, Oct 13 2022

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

A378295 Prime norms of ideals in Q(sqrt(10), sqrt(26)).

Original entry on oeis.org

2, 5, 13, 37, 67, 79, 83, 163, 191, 197, 199, 227, 293, 307, 311, 317, 397, 439, 521, 557, 569, 587, 599, 601, 613, 641, 643, 683, 719, 733, 751, 773, 787, 809, 827, 853, 877, 881, 911, 919, 947, 991, 1031, 1039, 1049, 1123, 1163, 1231, 1237, 1249, 1307, 1361, 1373, 1439, 1481, 1493
Offset: 1

Views

Author

Jovan Radenkovicc, Nov 22 2024

Keywords

Comments

Except for 2, 5 and 13, primes congruent to 1, 9, 37, 49, 67, 79, 81, 83, 93, 121, 123, 129, 159, 163, 187, 191, 197, 199, 203, 209, 213, 227, 231, 253, 267, 289, 293, 307, 311, 317, 321, 323, 329, 333, 357, 361, 391, 397, 399, 427, 437, 439, 441, 453, 471, 483, 511, 519 mod 520.
Primes in A378294.
Every prime p occurs in exactly one or all three of the sequences A038879, A038899 and A038945. This sequence lists the primes appearing in all three sequences.

Crossrefs

Cf. A038879, A038899, A038945, A378294.

Programs

  • Magma
    [p: p in PrimesUpTo(1500) | p in {2, 5, 13} or p mod 520 in [1, 9, 37, 49, 67, 79, 81, 83, 93, 121, 123, 129, 159, 163, 187, 191, 197, 199, 203, 209, 213, 227, 231, 253, 267, 289, 293, 307, 311, 317, 321, 323, 329, 333, 357, 361, 391, 397, 399, 427, 437, 439, 441, 453, 471, 483, 511, 519]];
Showing 1-3 of 3 results.

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

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