Primitive binary trinomial middle exponents.

[Ed Pegg]

Would this be OEIS-worthy?

Some initial minimal low-weight binary primitive polynomials are 
1+x+x^2, 1+x+x^3, 1+x+x^4, 1+x^2+x^5, 1+x+x^6, 1+x+x^7, 1+x^2+x^3+x^4+x^8, 1+x^4+x^9, 1+x^3+x^10, 1+x^2+x^11, 1+x+x^4+x^6+x^12  

If a trinomial exists, give the lowest middle exponent. Otherwise give zero.  

{1, 1, 1, 2, 1, 1, 0, 4, 3, 2, 0, 0, 0, 1, 0, 3, 7, 0, 3, 2, 1, 5,
0, 3, 0, 0, 3, 2, 0, 3, 0, 13, 0, 2, 11, 0, 0, 4, 0, 3, 0, 0, 0, 0,
0, 5, 0, 9, 0, 0, 3, 0, 0, 24, 0, 7, 19, 0, 1, 0, 0, 1, 0, 18, 0,
0, 9, 0, 0, 6, 0, 25, 0, 0, 0, 0, 0, 9, 0, 4, 0, 0, 13, 0, 0, 13,
0, 38, 0, 0, 0, 2, 21, 11, 0, 6, 11, 0, 37, 0, 0, 9, 0, 16, 15, 0,
31, 0, 0, 10, 0, 9, 0, 0, 0, 0, 33, 8, 0, 18, 0, 2, 37, 0, 0, 1, 0,
5, 3, 0, 29, 0, 57, 11, 0, 21, 0, 0, 29, 0, 21, 0, 0, 52, 0, 0, 27,
0, 53, 3, 0, 1, 0, 0, 0, 0, 0, 31, 0, 18, 0, 0, 0, 0, 0, 6, 0, 34,
23, 0, 7, 0, 13, 6, 0, 8, 87, 0, 0, 0, 0, 56, 0, 24, 0, 0, 0, 0, 0,
9, 0, 15, 87, 0, 0, 0, 65, 34, 0, 14, 55, 0, 0, 0, 0, 43, 0, 6, 0,
0, 105, 0, 0, 23, 0, 45, 11, 0, 0, 0, 0, 33, 0, 32, 0, 0, 0, 0, 0,
26, 0, 74, 31, 0, 5, 0, 0, 36, 0, 70, 0, 0, 0, 0, 0, 82, 0, 86, 103,
0, 67, 0, 0, 52, 0, 12, 83, 0, 0, 0, 0, 93, 0, 42, 47, 0, 25, 0, 53,
58, 0, 23, 67, 0, 0, 0, 5, 5, 0, 93, 35, 0, 119, 0, 69, 71, 0, 21,
0, 0, 97, 0, 61, 48, 0, 5, 0, 0, 7, 0, 41, 0, 0, 102, 0, 0, 0, 0, 0,
0, 0, 79, 15, 0, 135, 0, 0, 36, 0, 31, 67, 0, 0, 0, 0, 34, 0, 50, 0,
0, 123, 2, 0, 0, 0, 55, 0, 0, 0, 0, 125, 75, 0, 22, 0, 0, 0, 0, 53,
34, 0, 69, 0, 0, 0, 0, 0, 68, 0, 0, 63, 0, 67, 0, 29, 21, 0, 91,
139, 0, 0, 0, 0, 16, 0, 41, 43, 0, 47, 0, 81, 90, 0, 6, 83, 0, 0, 0,
89, 28, 0, 7, 135, 0, 25, 0, 0, 86, 0, 152, 0, 0, 189, 0, 157, 71,
0, 87, 0, 0, 147, 0, 0, 102, 0, 107, 0, 0, 0, 0, 149, 25, 0, 12, 0,
0, 105, 0, 0, 120, 0, 33, 0, 0, 165, 0, 65, 49, 0, 31, 0, 0, 0, 0,
105, 73, 0, 134, 79, 0, 0, 0, 0, 38, 0, 16, 203, 0, 61, 0, 73, 93,
0, 59, 0, 0, 0, 0, 149, 1, 0, 0, 191, 0, 15, 0, 121, 104, 0, 138, 0,
0, 105, 0, 0, 94, 0, 83, 219, 0, 0, 0, 137, 76, 0, 78, 0, 0, 0, 0,
0, 3, 0, 156, 95, 0, 109, 0, 0, 10, 0, 85, 0, 0, 0, 0, 33, 79, 0, 32,
0, 0, 167, 0, 0, 47, 0, 42, 0, 0, 1, 0, 0, 0, 0, 94, 0, 0, 179, 0,
0, 16, 0, 122, 0, 0, 0, 0, 193, 135, 0, 39, 0, 0, 153, 0, 0, 34, 0,
71, 0, 0, 163, 0, 153, 143, 0, 77, 67, 0, 0, 0, 13, 146, 0, 25, 0,
0, 0, 0, 85, 130, 0, 121, 0, 0, 151, 0, 93, 0, 0, 86, 19, 0, 0, 0,
0, 30, 0, 201, 0, 0, 0, 0, 0, 105, 0, 31, 127, 0, 0, 0, 0, 211, 0,
200, 0, 0, 0, 0, 297, 68, 0, 133, 0, 0, 223, 0, 0, 307, 0, 101, 315,
0, 0, 0, 0, 16, 0, 11, 119, 0, 0, 0, 249, 5, 0, 37, 3, 0, 93, 0, 0,
88, 0, 38, 55, 0, 0, 0, 297, 257, 0, 33, 0, 0, 0, 0, 153, 15, 0, 28,
0, 0, 241, 0, 0, 66, 0, 0, 0, 0, 0, 0, 197, 13, 0, 14, 0, 0, 299, 0,
0, 212, 0, 267, 215, 0, 0, 0, 37, 0, 0, 19, 0, 0, 287, 0, 0, 92, 0,
41, 23, 0, 183, 0, 0, 150, 0, 9, 231, 0, 0, 0, 5, 180, 0, 58, 147,
0, 0, 0, 0, 44, 0, 5, 347, 0, 153, 0, 0, 90, 0, 258, 351, 0, 0, 0,
0, 18, 0, 158, 19, 0, 349, 0, 0, 98, 0, 3, 83, 0, 0, 0, 0, 168, 0,
120, 0, 0, 7, 0, 185, 367, 0, 29, 375, 0, 0, 0, 329, 68, 0, 92, 0,
0, 0, 0, 0, 30, 0, 253, 143, 0, 0, 0, 0, 25, 0, 217, 0, 0, 295, 0,
141, 7, 0, 15, 299, 0, 167, 0, 145, 333, 0, 52, 119, 0, 0, 0, 0, 9,
0, 38, 255, 0, 205, 0, 0, 49, 0, 149, 0, 0, 0, 0, 61, 54, 0, 144,
47, 0, 0, 2, 0, 136, 0, 253, 111, 0, 0, 0, 0, 29, 0, 119, 0, 0, 0,
0, 349, 0, 0, 1, 75, 0, 145, 0, 0, 378, 0, 0, 0, 0, 0, 0, 0, 11, 0,
78, 0, 0, 173, 0, 0, 147, 0, 169, 0, 0, 31, 0, 173, 12, 0, 113, 207,
0, 1, 0, 0, 160, 0, 117, 187, 0, 143, 0, 0, 204, 0, 91, 0, 0, 0, 0,
77, 36, 0, 221, 0, 0, 0, 0, 365, 403, 0, 0, 0, 0, 275, 0, 0, 417, 0,
217, 207, 0, 0, 0, 0, 24, 0, 79, 0, 0, 0, 0, 0, 260, 0, 168, 0, 0,
305, 0, 0, 143, 0, 18, 0, 0, 103, 0, 0, 36, 0, 74, 0, 0, 115, 0, 0,
19, 0, 15, 0, 0, 0, 0, 277, 230, 0, 222, 0, 0, 121, 0, 0, 39, 0, 62,
223, 0, 0, 0, 101, 59, 0, 17, 0, 0, 0, 0, 0, 75, 0, 55, 0, 0, 0, 0,
385, 186, 0, 0, 0, 0, 461, 0, 317, 7, 0, 294, 35, 0, 203, 0, 93, 68,
0, 108, 75, 0, 411, 0, 0, 21, 0, 412, 439, 0, 41, 0, 0, 10, 0, 141,
0, 0, 291, 0, 105, 24, 0, 198, 27, 0, 0, 0, 0, 168, 0, 463, 0, 0,
0, 0, 0, 50, 0, 0, 0, 0, 0, 0, 445, 230, 0, 24, 407, 0, 189, 0, 0,
112, 0, 91, 79, 0, 23, 0, 261, 139, 0, 14, 83, 0, 0, 0, 117, 65, 0,
21, 195, 0, 327, 0, 0, 13, 0, 107, 0, 0, 479, 0, 0, 283, 0, 62, 0,
0, 0, 0, 309, 27, 0, 103, 551, 0, 0, 0, 0, 9, 0, 277, 31, 0, 539,
0, 357, 0, 0, 227, 131, 0, 23, 0, 0, 90, 0, 241, 75, 0, 307, 0,
245, 66, 0, 365, 0, 0, 19, 0, 189, 133, 0, 114, 0, 0, 0, 0, 133,
476, 0, 16, 375, 0, 0, 0, 0, 87, 0, 134, 171, 0, 413, 0, 233, 196,
0, 173, 0, 0, 519, 0, 0, 114, 0}

Related:  https://oeis.org/A073639  

[Neil Sloane]

> Would this be OEIS-worthy?

Yes!

I think we probably already have the n for which x^n + x^i + 1 is primitive for some i, so that should be a cross-ref.

Best regards

Neil 

Neil J. A. Sloane, Chairman, OEIS Foundation.

Also Visiting Scientist, Math. Dept., Rutgers University, 

Email: njas...@gmail.com

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[Ed Pegg]

https://oeis.org/A073726  
Primitive irreducible trinomials: x^n + x^k + 1 is a primitive irreducible polynomial (mod 2) for some k with 0 < k < n.

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[Ed Pegg]

Minimal k  for  https://oeis.org/A073726   
1, 1, 1, 2, 1, 1, 4, 3, 2, 1, 3, 7, 3, 2, 1, 5, 3, 3, 2, 3, 13, 2, 11, 4, 3, 5, 9, 3, 24, 7, 19, 1, 1, 18, 9, 6, 25, 9, 4, 13, 13, 38, 2, 21, 11, 6, 11, 37, 9, 16, 15, 31, 10, 9, 33, 8, 18, 2, 37, 1, 5, 3, 29, 57, 11, 21, 29, 21, 52, 27, 53, 3, 1, 31, 18, 6, 34, 23, 7, 13, 6, 8, 87, 56, 24, 9, 15, 87, 65, 34, 14, 55

[Ed Pegg]

I should have looked it up...  https://oeis.org/A074744  lists the k values.