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A163304
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n^4 + 984n^3 + 902n^2 + 394n + 858.
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2
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858, 3139, 13142, 36807, 80098, 149003, 249534, 387727, 569642, 801363, 1088998, 1438679, 1856562, 2348827, 2921678, 3581343, 4334074, 5186147, 6143862, 7213543, 8401538, 9714219, 11157982, 12739247, 14464458, 16340083, 18372614, 20568567, 22934482, 25476923
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Comment from Vincenco Librandi, Aug 23 2011 (Start):
This deals with primitive polynomials in GF_k(p). There are p^k monic k-th order polynomials J(p) = x^k + a(k-1)*x^(k-1) + ... + a(0), because there are k independent coefficients a(.), each restricted modulo the prime p. phi(p^k-1)/k of these polynomials are primitive, where phi=A000010. [Example for p=7 and k=2: phi(7^2-1)/2 = phi(48)/2 = 16/2=8. See A011260 for p=2, A027385 for p=3, A027741 for p=5 etc.] Of these sets of primitive polynomials we select with p=1031 the polynomial x^3+73*x^2+x+67 for k=3 in A163303 and x^4+984*x^3+90*x^2+394-x+858 for k=4 in A163304 by the following criteria (This could be extended to k=5, 6,...): Let r = (p^k -1)/(p-1). We demand (see Theorem 1 in Hansen-Mullen)
i) (-1)^k a(0) is a primitive element of J(p).
ii) The remainder of the division of x^r through the polynomial equals (-1)^k a(0).
iii) The remainder of the division of x^(r/q) through the polynomial must have positive degree for each prime divisor q|r.
(End)
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REFERENCES
| Marco Cugiani, Metodi numerico statistici (Collezione di Matematica applicata n.7), UTET Torino, 1980, pp.78-84
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index to sequences with linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
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FORMULA
| G.f.: (858-1151*x+6027*x^2-6093*x^3+383*x^4)/(1-x)^5. - Bruno Berselli, Aug 24 2011
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MATHEMATICA
| Table[n^4+984n^3+902n^2+394n+858, {n, 0, 30}] (* From Harvey P. Dale, Aug 16 2011 *)
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PROG
| (MAGMA) [n^4+984*n^3+902*n^2+394*n+858: n in [0..30]]; Vincenzo Librandi, Aug 17 2011
(PARI) a(n) = n^4+984*n^3+902*n^2+394*n+858 \\ Charles R Greathouse IV, Aug 17 2011
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CROSSREFS
| Cf. A163303.
Sequence in context: A198527 A087002 A046394 * A183628 A187281 A108822
Adjacent sequences: A163301 A163302 A163303 * A163305 A163306 A163307
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KEYWORD
| nonn,easy
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AUTHOR
| Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jul 24 2009
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EXTENSIONS
| Corrected and extended by Harvey P. Dale, Aug 16 2011
Offset changed from 1 to 0 by Vincenzo Librandi, Aug 17 2011
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