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 A278572 Irregular triangle read by rows: row n lists values of k in range 1 <= k <= n/2 such x^n + x^k + 1 is irreducible (mod 2), or -1 if no such k exists. 2
 1, 1, 1, 2, 1, 3, 1, 3, -1, 1, 4, 3, 2, 3, 5, -1, 5, 1, 4, 7, -1, 3, 5, 6, 3, 7, 9, -1, 3, 5, 2, 7, 1, 5, 9, -1, 3, 7, -1, -1, 1, 3, 9, 13, 2, 1, 9, 3, 6, 7, 13, -1, 10, 13, 7, 2, 9, 11, 15, -1, -1, 4, 8, 14, -1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,4 COMMENTS This is the format used by John Brillhart (1968) and Zierler and Brillhart (1968). REFERENCES Alanen, J. D., and Donald E. Knuth. "Tables of finite fields." Sankhyā: The Indian Journal of Statistics, Series A (1964): 305-328. John Brillhart, On primitive trinomials (mod 2), unpublished Bell Labs Memorandum, 1968. Marsh, Richard W. Table of irreducible polynomials over GF (2) through degree 19. Office of Technical Services, US Department of Commerce, 1957. Mossige, S. "Table of irreducible polynomials over 𝐺𝐹[2] of degrees 10 through 20." Mathematics of Computation 26.120 (1972): 1007-1009. LINKS Robert Israel, Table of n, a(n) for n = 2..4328 (rows 2 to 2170, flattened) Joerg Arndt, Complete list of primitive trinomials over GF(2) up to degree 400. (Lists primitive trinomials only.) Joerg Arndt, Complete list of primitive trinomials over GF(2) up to degree 400 [Cached copy, with permission] R. P. Brent, Trinomial Log Files and Certificates A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see Table 4.6. N. Zierler and J. Brillhart, On primitive trinomials (mod 2), Information and Control 13 1968 541-554. N. Zierler and J. Brillhart, On primitive trinomials (mod 2), II, Information and Control 14 1969 566-569. EXAMPLE Triangle begins: 1, 1, 1, 2, 1, 3, 1, 3, -1, 1, 4, 3, 2, 3, 5, -1, 5, 1, 4, 7, -1, 3, 5, 6, ... MAPLE T:= proc(n) local L; L:= select(k -> Irreduc(x^n+x^k+1) mod 2, [\$1..n/2]); if L = [] then -1 else op(L) fi end proc: map(T, [\$2..100]); # Robert Israel, Mar 28 2017 MATHEMATICA DeleteCases[#, 0] & /@ Table[Boole[IrreduciblePolynomialQ[x^n + x^# + 1, Modulus -> 2]] # & /@ Range[Floor[n/2]], {n, 2, 40}] /. {} -> {-1} // Flatten (* Michael De Vlieger, Mar 28 2017 *) CROSSREFS Cf. A001153, A057646, A057774, A073571, A073646, A073726, A074743, A278573. Rows n that contain particular numbers: 1 (A002475), 2 (A057460), 3 (A057461), 4 (A057463), 5 (A057474), 6 (A057476), 7 (A057477), 8 (A057478), 9 (A057479), 10 (A057480), 11 (A057481), 12 (A057482), 13 (A057483). Sequence in context: A046924 A015710 A108415 * A136644 A111963 A206923 Adjacent sequences:  A278569 A278570 A278571 * A278573 A278574 A278575 KEYWORD sign,tabf,more AUTHOR N. J. A. Sloane, Nov 27 2016 STATUS approved

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Last modified February 21 03:06 EST 2019. Contains 320364 sequences. (Running on oeis4.)