The number of irreducible 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	3	0	6	0
7	0	0	0	4	0	10	0	4
8	0	0	0	0	0	17	0	13	0
9	0	0	0	4	0	22	0	28	0	2
10	0	0	0	2	0	38	0	44	0	14	0
11	0	0	0	2	0	46	0	84	0	52	0	2
12	0	0	0	4	0	54	0	152	0	110	0	14	0
13	0	0	0	0	0	66	0	236	0	264	0	60	0	4
14	0	0	0	2	0	73	0	357	0	500	0	214	0	15	0
15	0	0	0	6	0	98	0	546	0	898	0	546	0	82	0	6
16	0	0	0	0	0	94	0	734	0	1587	0	1304	0	337	0	24	0
17	0	0	0	6	0	152	0	1050	0	2674	0	2696	0	1006	0	122	0	4
18	0	0	0	5	0	124	0	1374	0	4316	0	5406	0	2745	0	531	0	30	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	4	0	199	0	2325	0	9995	0	18405	0	15227	0	5368	0	815	0	39	0
21	0	0	0	4	0	184	0	2892	0	14988	0	31848	0	32144	0	14698	0	2888	0	212	0	0
22	0	0	0	2	0	226	0	3650	0	20993	0	53602	0	64163	0	36877	0	9928	0	1078	0	38	0
23	0	0	0	4	0	296	0	4660	0	29458	0	86626	0	122502	0	86528	0	29748	0	4606	0	286	0	8
24	0	0	0	0	0	202	0	5191	0	40861	0	136378	0	225569	0	190357	0	81708	0	17063	0	1509	0	32	0
25	0	0	0	4	0	406	0	6938	0	55202	0	208988	0	399576	0	399560	0	208542	0	55752	0	6880	0	324	0	4
26	0	0	0	0	0	328	0	8012	0	74404	0	314185	0	685607	0	799042	0	503547	0	166341	0	27390	0	1899	0	40	0
27	0	0	0	0	0	334	0	9576	0	97266	0	463084	0	1143902	0	1541160	0	1144494	0	464038	0	97262	0	9512	0	380	0	0
28	0	0	0	8	0	418	0	11476	0	126779	0	669932	0	1864232	0	2862833	0	2482205	0	1206184	0	317395	0	42358	0	2536	0	38	0
29	0	0	0	2	0	380	0	13000	0	162742	0	953474	0	2966518	0	5158444	0	5163764	0	2963748	0	953582	0	163112	0	13560	0	460	0	4
30	0	0	0	4	0	486	0	15842	0	207089	0	1333022	0	4619193	0	9055093	0	10333339	0	6917423	0	2670580	0	571435	0	63632	0	3071	0	58	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 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 for primitive polynomials is also available.
• Created by Frank Ruskey. Please let me know if you make use of this table.