

A278573


Irregular triangle read by rows: row n lists values of k in range 1 <= k <= n1 such x^n + x^k + 1 is irreducible (mod 2), or 1 if no such k exists.


2



1, 1, 2, 1, 3, 2, 3, 1, 3, 5, 1, 3, 4, 6, 1, 1, 4, 5, 8, 3, 7, 2, 9, 3, 5, 7, 9, 1, 5, 9, 1, 4, 7, 8, 11, 14, 1, 3, 5, 6, 11, 12, 14, 3, 7, 9, 11, 15, 1, 3, 5, 15, 17, 2, 7, 14, 19, 1, 21, 5, 9, 14, 18, 1, 3, 7, 18, 22, 1, 1, 1, 3, 9, 13, 15, 19, 25, 27, 2, 27, 1, 9, 21, 29, 3, 6, 7, 13
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OFFSET

2,3


COMMENTS

Row n (if it is not 1) is invariant under the map k > nk.  Robert Israel, Mar 14 2018


REFERENCES

Alanen, J. D., and Donald E. Knuth. "Tables of finite fields." Sankhyā: The Indian Journal of Statistics, Series A (1964): 305328.
John Brillhart, On primitive trinomials (mod 2), unpublished Bell Labs Memorandum, 1968.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): 10071009.


LINKS

Robert Israel, Table of n, a(n) for n = 2..4558 (rows 2 to 1300, 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 541554.
N. Zierler and J. Brillhart, On primitive trinomials (mod 2), II, Information and Control 14 1969 566569.
Index entries for sequences related to trinomials over GF(2)


EXAMPLE

Triangle begins:
1,
1, 2,
1, 3,
2, 3,
1, 3, 5,
1, 3, 4, 6,
1,
1, 4, 5, 8,
3, 7,
2, 9,
3, 5, 7, 9,
1,
5, 9,
1, 4, 7, 8, 11, 14,
1,
3, 5, 6, 11, 12, 14,
3, 7, 9, 11, 15,
1,
3, 5, 15, 17,
2, 7, 14, 19,
1, 21,
...


MAPLE

for n from 2 to 30 do
S:= select(k > Irreduc(x^n+x^k+1) mod 2, [$1..n1]);
if S = [] then print(1) else print(op(S)) fi
od: # Robert Israel, Mar 14 2018


CROSSREFS

Cf. A001153, A057646, A057774, A073571, A073646, A073726, A074743, A278572.
Sequence in context: A151682 A318928 A159918 * A108663 A057940 A097285
Adjacent sequences: A278570 A278571 A278572 * A278574 A278575 A278576


KEYWORD

sign,tabf


AUTHOR

N. J. A. Sloane, Nov 27 2016


STATUS

approved



