The number of primitive polynomials over GF(2)
with a given number of non-zero coefficients.

	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	0	1
2	0	0	0
3	0	0	0	2
4	0	0	0	2	0
5	0	0	0	2	0	4
6	0	0	0	2	0	4	0
7	0	0	0	4	0	10	0	4
8	0	0	0	0	0	12	0	4	0
9	0	0	0	2	0	16	0	28	0	2
10	0	0	0	2	0	20	0	28	0	10	0
11	0	0	0	2	0	44	0	80	0	50	0	0
12	0	0	0	0	0	18	0	86	0	36	0	4	0
13	0	0	0	0	0	66	0	236	0	264	0	60	0	4
14	0	0	0	0	0	42	0	226	0	338	0	140	0	10	0
15	0	0	0	6	0	82	0	470	0	720	0	450	0	66	0	6
16	0	0	0	0	0	52	0	368	0	812	0	648	0	156	0	12	0
17	0	0	0	6	0	152	0	1050	0	2674	0	2696	0	1006	0	122	0	4
18	0	0	0	2	0	72	0	718	0	2296	0	2910	0	1470	0	284	0	24	0
19	0	0	0	0	0	158	0	1774	0	6696	0	10238	0	6766	0	1772	0	190	0	0
20	0	0	0	2	0	100	0	1104	0	4522	0	8436	0	7000	0	2460	0	354	0	22	0
21	0	0	0	2	0	164	0	2490	0	12628	0	27080	0	27278	0	12422	0	2442	0	166	0	0
22	0	0	0	2	0	122	0	2260	0	13218	0	33886	0	40362	0	23170	0	6288	0	698	0	26	0
23	0	0	0	4	0	292	0	4552	0	28820	0	84774	0	119786	0	84800	0	29140	0	4504	0	280	0	8
24	0	0	0	0	0	94	0	2110	0	16346	0	53832	0	89104	0	75210	0	32424	0	6720	0	628	0	12	0
25	0	0	0	4	0	396	0	6702	0	53386	0	201742	0	385742	0	385822	0	201424	0	53806	0	6662	0	310	0	4
26	0	0	0	0	0	214	0	5344	0	49470	0	209464	0	457532	0	532238	0	335042	0	110940	0	18380	0	1252	0	24	0
27	0	0	0	0	0	286	0	8112	0	82292	0	391440	0	967040	0	1302862	0	967606	0	392360	0	82174	0	8002	0	322	0	0
28	0	0	0	6	0	188	0	5816	0	62804	0	331652	0	920970	0	1417026	0	1228258	0	596122	0	156716	0	20834	0	1222	0	18	0
29	0	0	0	2	0	378	0	12910	0	161768	0	948030	0	2949628	0	5129024	0	5134762	0	2947238	0	947916	0	162204	0	13486	0	458	0	4
30	0	0	0	0	0	208	0	7988	0	102716	0	665442	0	2300670	0	4508866	0	5142250	0	3444108	0	1329736	0	285058	0	31486	0	1444	0	28	0
31	0	0	0	8	0	584	0	18810	0	261958	0	1847144	0	7053018	0	15449530	0	20011118	0	15456044	0	7049276	0	1845032	0	262214	0	18368	0	554	0	8

Notes:

• The primitive 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.
• Note that all entries are even. This is true since the reciprocal of a primitive polynomial is also primitive, the reciprocal of a polynomial has the same density, and no primitive polynomial is self-reciprocal.
• The corresponding table for irreducible polynomials is also available.
• Created by Frank Ruskey. Please let me know if you make use of this table.