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A047363 Numbers that are congruent to {0, 2, 3, 4, 5} mod 7. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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: A039055 A047504 A088962 * A191917 A228894 A325367

Adjacent sequences:  A047360 A047361 A047362 * A047364 A047365 A047366

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 19 11:09 EST 2019. Contains 329319 sequences. (Running on oeis4.)