A047363 Numbers that are congruent to {0, 2, 3, 4, 5} mod 7.

0, 2, 3, 4, 5, 7, 9, 10, 11, 12, 14, 16, 17, 18, 19, 21, 23, 24, 25, 26, 28, 30, 31, 32, 33, 35, 37, 38, 39, 40, 42, 44, 45, 46, 47, 49, 51, 52, 53, 54, 56, 58, 59, 60, 61, 63, 65, 66, 67, 68, 70, 72, 73, 74, 75, 77, 79

Offset

1,2

Comments

Conjecture: Apart from 0, and the further exclusions noted below, the sequence gives the values of c/6 such that an infinite number of primes, p, result in both p^3+c and p^3-c being positive primes. Taking the complement we say: the excluded c/6 values are {1,6} mod 7. See A005097 for a conjecture on the modulo patterns of excluded c/6 values for the general case of p^q + c and p^q - c both prime, for any q > 0, and see A047222 for q=2. Note that polynomial factorization also excludes a few c/6 values. This occurs here when c is an even cube (A016743), which requires a further exclusion of certain c/6 values in this sequence when (6c)^3/6 == 0 (mod 7), or c/6 = {0, 12348, 98784, ...}. - Richard R. Forberg, Jun 28 2016

Links

Table of n, a(n) for n=1..57.

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

Formula

G.f.: x^2*(2*x^2 + 3*x + 2)*(x^2 - x + 1) / ( (x^4 + x^3 + x^2 + x + 1)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011

a(n) = a(n-1) + a(n-5) - a(n-6). - Wesley Ivan Hurt, Sep 03 2022

Mathematica

Table[7 n + {0, 2, 3, 4, 5}, {n, 0, 12}] // Flatten (* or *)
Select[Range[0, 79], ! MemberQ[{1, 6}, Mod[#, 7]] &] (* or *)
Rest@ CoefficientList[Series[x^2 (2 x^2 + 3 x + 2) (x^2 - x + 1)/((x^4 + x^3 + x^2 + x + 1) (x - 1)^2), {x, 0, 57}], x] (* Michael De Vlieger, Jul 25 2016 *)

Crossrefs

Cf. A047222, A005097.

Sequence in context: A047504 A359830 A088962 * A191917 A228894 A325367

Adjacent sequences: A047360 A047361 A047362 * A047364 A047365 A047366

Keyword

nonn

Author

N. J. A. Sloane

Status

approved