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.

A349202 Numbers k of the form (x + y)*(x^2 + y^2) such that (x + y) and (x^2 + y^2) are primes.

Original entry on oeis.org

4, 15, 65, 85, 203, 259, 671, 803, 1111, 1157, 1261, 1417, 2533, 2669, 3439, 3667, 3893, 4369, 4579, 5567, 6187, 6371, 8027, 9407, 12209, 12557, 13369, 16151, 16771, 17429, 18383, 18589, 20491, 21257, 21731, 26233, 28453, 29489, 30673, 34973, 36121, 36889
Offset: 1

Views

Author

Peter Luschny, Nov 11 2021

Keywords

Examples

			1157 is in this sequence because 1157 = (5 + 8)*(5^2 + 8^2) = 13*89.

Links

Crossrefs

Cf. A001358, A348897.

Programs

  • Julia
    # Returns the terms less than or equal to b^3.
using Nemo
function A349202List(b)
    b3 = b^3; R = Int[]
    for n in 1:b
        for k in 0:n
            a = (n + k) * (n^2 + k^2)
            a > b3 && break
            isprime(n+k) && isprime(n^2 + k^2) && push!(R, a)
    end end
sort(R) end
A349202List(34) |> println

Formula

Intersection of A001358 and A348897.

A349200 Loeschian numbers of the form (x + y)*(x^2 + y^2).

Original entry on oeis.org

0, 1, 4, 27, 64, 108, 156, 175, 256, 259, 343, 400, 729, 1261, 1372, 1417, 1728, 1875, 2197, 2916, 3439, 3492, 3667, 4096, 4212, 4579, 4725, 6175, 6859, 6912, 6993, 7104, 7825, 8112, 8125, 8425, 8788, 9261, 9264, 9325, 9925, 9984, 10800, 11200, 11425, 11712
Offset: 1

Views

Author

Peter Luschny, Nov 10 2021

Keywords

Comments

k is in this sequence if there exist numbers x, y, v, w such that k = x^2 + x*y + y^2 = (v + w)*(v^2 + w^2). We call (x, y, v, w) a witness of k. k can have different witnesses.

Examples

			729  = 27^2 + 27*0 + 0^2   = (9 + 0)*(9^2 + 0^2).
3492 = 48^2 + 48*18 + 18^2 = (13 + 5)*(13^2 + 5^2).
3667 = 53^2 + 53*13 + 13^2 = (12 + 7)*(12^2 + 7^2).

Links

Crossrefs

Cf. A003136, A348897, A349202.

Programs

  • Julia
    # Returns the terms less than or equal to b^3.
# Uses the function isA003136 from A003136.
function A349200List(b)
    b3 = b^3; R = [0]
    for n in 1:b
        for k in 0:n
            a = (n + k) * (n^2 + k^2)
            a > b3 && break
            isA003136(a) && push!(R, a)
    end end
sort(R) end
A349200List(24) |> println

Formula

Intersection of A003136 and A348897.

A349201 a(n) = [x^n] ((x^2*(1 + 3*x + x^2 - 2*x^3 + 3*x^4 + x^5 - x^6))/((-1 + x)^4 *(1 + x)^3)).

Original entry on oeis.org

0, 1, 4, 8, 15, 27, 40, 64, 85, 125, 156, 216, 259, 343, 400, 512, 585, 729, 820, 1000, 1111, 1331, 1464, 1728, 1885, 2197, 2380, 2744, 2955, 3375, 3616, 4096, 4369, 4913, 5220, 5832, 6175, 6859, 7240, 8000, 8421, 9261, 9724, 10648, 11155, 12167, 12720, 13824
Offset: 1

Views

Author

Peter Luschny, Nov 10 2021

Keywords

Comments

A subsequence of A348897, i.e. each term of this sequence has the form (x + y)*(x^2 + y^2).

Links

Crossrefs

Cf. A348897.

Programs

  • Mathematica
    Join[{0},LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,4,8,15,27,40,64},47]] (* Stefano Spezia, Nov 11 2021 *)

Formula

From Stefano Spezia, Nov 11 2021: (Start)
a(n) = ((5 + 3*n - n^2)*(1 - (-1)^n) + 2*n^3)/16 for n > 1.
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4)+ 3*a(n-5) + a(n-6) - a(n-7) for n > 8. (End)
Showing 1-3 of 3 results.

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

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