A344146 a(n) is the number of pentanomials x^n + x^a + x^b + x^c + 1 that are irreducible over GF(2) for n > a > b > c > 0.

1, 4, 6, 10, 17, 22, 38, 46, 54, 66, 73, 98, 94, 152, 124, 158, 199, 184, 226, 296, 202, 406, 328, 334, 418, 380, 486, 584, 351, 666, 578, 658, 896, 604, 728, 964, 577, 1128, 925, 846, 1286, 898, 1102, 1520, 760, 1628, 1421, 1312, 1837, 1298, 1580, 2220, 1142, 2346, 1764, 1524, 2782

Offset: 4

Author

Jianing Song, May 10 2021

Keywords

• nonn
• hard

Comments

It is conjectured that a(n) > 0 for all n >= 4.

Examples

			a(4) = 1 because there is only one irreducible pentanomial of degree 4 over GF(2), namely x^4 + x^3 + x^2 + x + 1.
a(6) = 4 because there are 4 irreducible pentanomials of degree 6 over GF(2): x^6 + x^4 + x^2 + x + 1, x^6 + x^4 + x^3 + x + 1, x^6 + x^5 + x^2 + x + 1, x^6 + x^5 + x^3 + x^2 + 1, x^6 + x^5 + x^4 + x + 1 and x^6 + x^5 + x^4 + x^2 + 1.
a(7) = 10 since the 10 irreducible pentanomials of degree 6 over GF(2) are of the form x^7 + x^a + x^b + x^c + 1 for (a,b,c) = (3,2,1), (4,3,2), (5,2,1), (5,3,1), (5,4,3), (6,3,1), (6,4,1), (6,4,2), (6,5,2), (6,5,4).
		

Links

• Jianing Song, Table of n, a(n) for n = 4..173

Crossrefs

Cf. A014580 (irreducible polynomials over GF(2) encoded as binary numbers), A057646.

Programs

PARI

a(n) = sum(a=3, n-1, sum(b=2, a-1, sum(c=1, b-1, polisirreducible(Mod(x^n+x^a+x^b+x^c+1, 2)))))