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 1-2 of 2 results.

A141180 Primes of the form x^2+6*x*y-y^2 (as well as of the form 6*x^2+8*x*y+y^2).

Original entry on oeis.org

31, 41, 71, 79, 89, 151, 191, 199, 239, 241, 271, 281, 311, 359, 401, 409, 431, 439, 449, 479, 521, 569, 599, 601, 631, 641, 719, 751, 761, 769, 809, 839, 881, 911, 919, 929, 991, 1009, 1031, 1039, 1049, 1129, 1151, 1201, 1231, 1249, 1279, 1289, 1319, 1321, 1361, 1399, 1409, 1439
Offset: 1

Views

Author

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (lourdescm84(AT)hotmail.com), Jun 12 2008

Keywords

Comments

Discriminant = 40. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.
Also primes of form 10*u^2 - v^2. The transformation {u, v} = {-x, 3*x-y} yields the form in the title, and primes of form U^2 - 10*V^2, with transformation {U, V} = {x+3*y, y}. - Juan Arias-de-Reyna, Mar 19 2011
Therefore, these primes are composite in Q(sqrt(10)), as they can be factored thus: (-u + v*sqrt(10))*(u + v*sqrt(10)). - Alonso del Arte, Jul 22 2012
All primes p such that (p^2 - 1)/24 mod 10 = 0. See A024702. - Richard R. Forberg, Aug 27 2013

Examples

			a(2) = 41 because we can write 41 = 3^2 + 6*3*2 - 2^2 (or 41 = 6*2^2 + 8*2*1 + 1^2). Furthermore, notice that (-7 + 3*sqrt(10))(7 + 3*sqrt(10)) = 41.

References

  • Z. I. Borevich and I. R. Shafarevich, Number Theory.

Links

Crossrefs

Cf. A141179 (d=40) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65). A024702.
Cf. also A242664.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Programs

  • Mathematica
    Take[Select[Union[Flatten[Table[Abs[a^2 - 10b^2], {a, 0, 49}, {b, 0, 49}]]], PrimeQ], 50] (* Alonso del Arte, Jul 22 2012 *)
Select[Prime[Range[250]], MatchQ[Mod[#, 40], Alternatives[1, 9, 31, 39]]&] (* Jean-François Alcover, Oct 28 2016 *)

Extensions

Removed defective Mma program. - N. J. A. Sloane, Jun 06 2014

A370267 Numbers with an even number of prime factors not of the form 8m+-1 (counting repetitions).

Original entry on oeis.org

1, 4, 6, 7, 9, 10, 15, 16, 17, 22, 23, 24, 25, 26, 28, 31, 33, 36, 38, 39, 40, 41, 42, 47, 49, 54, 55, 57, 58, 60, 63, 64, 65, 68, 70, 71, 73, 74, 79, 81, 86, 87, 88, 89, 90, 92, 95, 96, 97, 100, 102, 103, 104, 105, 106, 111, 112, 113, 118, 119, 121, 122, 124, 127, 129
Offset: 1

Views

Author

Peter Munn, Feb 13 2024

Keywords

Comments

Construction by subgroup generation: (Start)
The set of numbers congruent to 1 modulo 8 (A017077) contains all the odd squares and generates a subgroup of the positive rational numbers (under multiplication) that contains no additional integers. The subgroup has an infinite number of cosets. The rest of the construction process extends the subgroup, reducing the number of cosets to 2, by choosing additional generators that are semiprime.
First we extend the subgroup to include all nonzero integer squares. As we already have the odd squares, we need only add 4, the square of the smallest prime, as a generator. The extended subgroup has only 8 cosets and its integer members are listed in A234000. To achieve a subgroup with 2 cosets we now add squarefree semiprime generators. The 2 smallest, 6 and 10, suffice.
The resulting subgroup has this sequence's terms as its integer members.
(End)
The equivalent process starting with numbers congruent to 1 modulo 3 (or 1 modulo 6) produces A189715. If we take its intersection with this sequence we get A370268, which starts with the first 72 nonzero numbers of the form x^2 + 6y^2 (see A002481). Similarly, if we start with numbers congruent to 1 modulo 5 (or 1 modulo 10) and take the resulting set's intersection with this sequence we get a set starting with the first 32 nonzero numbers of the form x^2 - 10y^2 (see A242664).
The construction process leads to a number of properties:
- The sequence is closed under multiplication and all integer ratios between terms are in the sequence.
- The sequence and its complement have the property that the terms of one can be generated by halving the even terms of the other. Each has asymptotic density 1/2.
Numbers whose squarefree part is congruent to {1,7} mod 8 or {6,10} mod 16.

Examples

			7 is prime, so 7 is its only prime factor, which has the form 8m-1. So 7 has an even number (zero) of prime factors not of the form 8m+-1, and therefore is in the sequence. In terms of the subgroup generators described at the start of the comments, (13*8+1) * 4 / (6*10) = 105 * 4/60 = 7.
110 = 2 * 5 * 11, so it has 3 prime factors and all 3 do not have the form 8m+-1. 3 is odd, so 110 is not in the sequence.

Crossrefs

Disjoint union of A004215, A055042, A055043 and A234000.
See the comments for the relationships with A002481, A017077, A189715, A242664, A370268.
Cf. A042999 (primes), A059897.

Programs

  • PARI
    isok(k) = {c = core(k); c%8 == 1 || c%8 == 7 || c%16 == 6 || c%16 == 10}
  • Python
    def A370267(n):
    def f(x): return n+x-sum(((y:=x>>(i<<1))-7>>3)+(y-1>>3)+2 for i in range((x.bit_length()>>1)+1))-sum(((z:=x>>(i<<1)+1)-5>>3)+(z-3>>3)+2 for i in range(x.bit_length()-1>>1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Mar 19 2025

Formula

{a(n) : n >= 1} = {A059897(i,j) : i in A234000, j in {1, 6, 10, 15}}.
Showing 1-2 of 2 results.

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

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