A344146 - OEIS

%I #17 Jun 26 2021 09:52:49

%S 1,4,6,10,17,22,38,46,54,66,73,98,94,152,124,158,199,184,226,296,202,

%T 406,328,334,418,380,486,584,351,666,578,658,896,604,728,964,577,1128,

%U 925,846,1286,898,1102,1520,760,1628,1421,1312,1837,1298,1580,2220,1142,2346,1764,1524,2782

%N 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.

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

%H Jianing Song, <a href="/A344146/b344146.txt">Table of n, a(n) for n = 4..173</a>

%e 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.

%e 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.

%e 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).

%o (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)))))

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

%K nonn,hard

%O 4,2

%A _Jianing Song_, May 10 2021