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A376932
a(n) is the index of the first GF(2)[X] polynomial that has n distinct irreducible factors.
0
1, 2, 6, 18, 166, 1806, 20382, 272706, 8323326, 158143194, 4319806194, 139715547110, 4563596609414, 154716297384250, 6051527318503338, 315946019303255670, 18477283150919171654, 1191953715632050834242, 76457609628854745786262, 4838004466153152832995822, 312401901306255000752991994, 20039165126917559409941672886
OFFSET
0,2
COMMENTS
a(n) is the least k such that A091221(k) = n.
EXAMPLE
a(3) = 18 because the 18th GF(2)[X] polynomial is X^4 + X = X * (X + 1) * (X^2 + X + 1) with 3 distinct irreducible factors over GF(2).
MAPLE
pol:= proc(x) local L, i;
L:= convert(x, base, 2);
add(L[i]*X^(i-1), i=1..nops(L));
end proc:
for m from 1 to 10 do
IP[m]:= select(t -> Irreduc(pol(t)) mod 2, [seq(x, x=2^m..2^(m+1)-1)]);
od:
nIP:= [seq(nops(IP[m]), m=1..10)]:
psnIP:= ListTools:-PartialSums(nIP):
f:= proc(n) local k, P0, r, xmin, x, i, s, P;
for k from 1 while n > psnIP[k] do od:
P0:= expand(mul(convert(map(pol, IP[i]), `*`), i=1..k-1)) mod 2;
if k = 1 then r:= n else r:= n - psnIP[k-1] fi;
xmin:= infinity;
for s in combinat:-choose(IP[k], r) do
P:= expand(P0 * mul(pol(i), i=s)) mod 2;
x:= eval(P, X=2);
xmin:= min(xmin, x);
od;
xmin
end proc:
seq(f(i), i=0..25);
CROSSREFS
Cf. A091221.
Sequence in context: A308549 A212653 A323104 * A208456 A056743 A064535
KEYWORD
nonn
AUTHOR
Robert Israel, Oct 11 2024
STATUS
approved