A047531 Numbers that are congruent to {2, 3, 7} mod 8.

2, 3, 7, 10, 11, 15, 18, 19, 23, 26, 27, 31, 34, 35, 39, 42, 43, 47, 50, 51, 55, 58, 59, 63, 66, 67, 71, 74, 75, 79, 82, 83, 87, 90, 91, 95, 98, 99, 103, 106, 107, 111, 114, 115, 119, 122, 123, 127, 130, 131, 135, 138, 139, 143, 146, 147, 151, 154, 155, 159

Offset

1,1

Comments

From Richard R. Forberg, Mar 14 2016 : (Start)

Numbers m such that p^2 + q^2 + r^2 - m is a square, for distinct odd primes p, q, and r. This follows from the fact that the sum of the squares of three odd primes is 3 mod 8, and squares are {0,1,4} mod 8. Changing -m to +m leads to A047438.

The expression p^2 + q^2 + r^2 - m can equal the square of an odd number only when m == 2 mod 8 (because all odd squares are 1 mod 8).

This same expression only equals squares of primes for m in {2, 10, 26, 34} mod 48, because when m = {18, 42} mod 48, the expression is congruent to {9, 33} mod 48, which matches the values of (6*k-3)^2 mod 48.

(End)

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Formula

From Chai Wah Wu, May 30 2016: (Start)

a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4.

G.f.: x*(x^3 + 4*x^2 + x + 2)/(x^4 - x^3 - x + 1). (End)

a(n) = (24*n + 5*sqrt(3)*sin(2*Pi*n/3) + 3*cos(2*Pi*n/3) - 12)/9. - Ilya Gutkovskiy, May 30 2016

a(3k) = 8k-1, a(3k-1) = 8k-5, a(3k-2) = 8k-6. - Wesley Ivan Hurt, Jun 10 2016

Maple

A047531:=n->(24*n-12+3*cos(2*n*Pi/3)+5*sqrt(3)*sin(2*n*Pi/3))/9: seq(A047531(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016

Mathematica

LinearRecurrence[{1, 0, 1, -1}, {2, 3, 7, 10}, 50] (* G. C. Greubel, May 30 2016 *)
Select[Range[200], MemberQ[{2, 3, 7}, Mod[#, 8]]&] (* Harvey P. Dale, Jan 18 2019 *)

Prog

(Magma) [n : n in [0..150] | n mod 8 in [2, 3, 7]]; // Wesley Ivan Hurt, Jun 10 2016

Crossrefs

Cf. A047438.

Sequence in context: A051637 A051471 A140794 * A102808 A215937 A140512

Adjacent sequences: A047528 A047529 A047530 * A047532 A047533 A047534

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved