The number of irreducible polynomials over GF(2)
of the form xⁿ+x^k+...

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
1	2
2	0	1
3	0	1	1
4	0	1	0	2
5	0	0	1	2	3
6	0	1	0	1	2	5
7	0	1	0	2	2	4	9
8	0	0	0	0	2	4	8	16
9	0	1	0	0	3	3	8	13	28
10	0	0	0	2	2	3	5	12	24	51
11	0	0	1	0	1	2	5	12	24	48	93
12	0	0	0	1	1	4	6	10	22	41	80	170
13	0	0	0	0	1	2	3	11	20	38	80	160	315
14	0	0	0	0	0	5	4	10	19	36	70	144	288	585
15	0	1	0	0	2	2	3	12	17	30	74	132	277	541	1091
16	0	0	0	0	0	4	2	4	18	36	56	120	256	512	1024	2048
17	0	0	0	2	0	4	4	7	16	32	62	120	244	484	960	1920	3855
18	0	0	0	1	0	2	3	8	14	31	53	110	222	469	900	1813	3626	7280
19	0	0	0	0	0	2	6	3	13	25	51	101	208	424	868	1728	3456	6912	13797
20	0	0	0	2	1	1	6	5	14	20	52	103	216	416	812	1616	3264	6579	13056	26214
21	0	0	1	0	0	2	4	3	18	24	48	90	193	376	784	1577	3104	6258	12458	24989	49929
22	0	1	0	0	0	3	1	5	11	20	51	95	187	379	759	1484	2980	5952	11880	23808	47616	95325
23	0	0	0	0	0	5	3	8	8	25	38	100	179	371	697	1443	2880	5684	11384	22832	45568	91136	182361
24	0	0	0	0	1	0	1	6	11	23	31	90	157	316	672	1388	2696	5368	10808	21952	43562	87508	174760	349520
25	0	0	0	2	0	1	0	4	11	22	43	92	167	340	635	1293	2611	5326	10468	21024	42032	83824	167731	335462	671088
26	0	0	0	0	1	0	3	6	11	21	35	83	166	314	620	1285	2521	5102	9968	20176	40408	80640	161200	322560	645120	1290555
27	0	0	0	0	0	1	1	4	11	20	39	67	144	307	612	1200	2383	4850	9726	19277	38968	77736	155119	310911	621224	1242904	2485504
28	0	1	0	2	0	0	3	6	6	17	41	78	163	290	588	1193	2316	4724	9318	18674	37476	75024	149600	299520	599040	1198665	2396160	4793490
29	0	0	1	0	2	0	0	3	8	14	48	69	130	269	555	1068	2238	4443	9143	18092	36216	72184	144552	288992	578304	1157344	2314240	4628480	9256395
30	0	1	0	0	0	0	2	5	10	24	29	61	138	269	534	1123	2183	4390	8751	17604	34916	69854	139866	279242	559066	1119026	2236553	4473647	8947294	17895679
31	0	0	0	2	0	2	3	5	9	14	35	74	121	277	522	1061	2109	4263	8516	17014	33708	67558	135354	270794	541456	1082544	2165024	4329536	8658944	17317888	34636833

Notes:

• The irreducible polynomial x is not counted anywhere in the above table; all others are counted.
• Row heading of row n is red if and only if 2ⁿ-1 is prime. In this case every irreducible polynomial is also primitive.
• The corresponding table form primitive polynomials is also available.
• Created by Frank Ruskey. Please let me know if you make use of this table.