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%I
%S 4,9,19,29,49,59,64,79,89,109,139,149,169,179,199,229,239,269,289,349,
%T 359,379,389,409,419,439,449,479,499,509,529,569,599,619,659,709,719,
%U 729,739,769,809,829,839,859,919,929,1009,1019,1024,1039,1049,1069,1109,1129,1229,1249
%N Primes powers (A246655) congruent to 4 mod 5.
%C Numbers k such that x^4 + x^3 + x^2 + x + 1 factors into two irreducible quadratic polynomials over GF(k).
%C Note that x^4 + x^3 + x^2 + x + 1 is reducible over GF(k) if and only if there exists some a in GF(k) such that a^2 - a - 1 = 0, and then x^4 + x^3 + x^2 + x + 1 = (x^2 + a*x + 1) * (x^2 + (1-a)*x + 1). There exists some a in GF(k) such that a^2 - a - 1 = 0 if and only if kronecker(k,5) = 1, or k == 1, 4 (mod 5). If k == 1 (mod 5), then x^4 + x^3 + x^2 + x + 1 can be further factored into four linear polynomials.
%C This sequence consists of numbers of the form p^(2e+1) where prime p == 4 (mod 5) and p^(4e+2) where prime p == 2, 3 (mod 5),
%H Marius A. Burtea, <a href="/A327753/b327753.txt">Table of n, a(n) for n = 1..10000</a>
%e k = 4: let GF(4) = GF(2)[w], w^2 + w + 1 = 0, then x^4 + x^3 + x^2 + x + 1 = (x^2 + w*x + 1)*(x^2 + (w+1)*x + 1);
%e k = 9: let GF(9) = GF(3)[i], i^2 = -1, then x^4 + x^3 + x^2 + x + 1 = (x^2 + (-1+i)*x + 1)*(x^2 + (-1-i)*x + 1);
%e k = 19: in GF(19), x^4 + x^3 + x^2 + x + 1 = (x^2 + 5x + 1)*(x^2 - 4x + 1).
%t Select[Range@ 1250, And[PrimePowerQ@ #, Mod[#, 5] == 4] &] (* _Michael De Vlieger_, Sep 27 2019 *)
%o (PARI) isok(n) = isprimepower(n) && (n%5==4)
%o (MAGMA) [n:n in [2..1250]|IsPrimePower(n) and (n mod 5 eq 4)]; // _Marius A. Burtea_, Sep 26 2019
%Y Cf. A137827, A327752.
%Y Intersection of A016897 and A246655.
%K nonn
%O 1,1
%A _Jianing Song_, Sep 24 2019
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