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A371554 Consider primitive pairs of integers (b, c) with b > 0 such that x^5 + b*x + c = 0 is irreducible and solvable by radicals: sequence gives values of c.

%I A371554 #13 Apr 20 2024 16:36:16
%S A371554 44,12,44,32,64,1344,576,1344,832,275,4170,2375,3750,4060,128700,
%T A371554 13243,1510620,24000,3348800,8788,467961,51072,133440,474214,61500,
%U A371554 128700,85683,514098,509197,199927,24000,3720000,21376538,210990,486343,114244,12681084
%N A371554 Consider primitive pairs of integers (b, c) with b > 0 such that x^5 + b*x + c = 0 is irreducible and solvable by radicals: sequence gives values of c.
%C A371554 See A371553.
%H A371554 Ben Whitmore, <a href="/A371554/b371554.txt">Table of n, a(n) for n = 1..95</a>
%F A371554 x^5 + A371553(n)*x + a(n) is irreducible and solvable by radicals.
%e A371554 a(1) = 44 because A371553(1) = 11, and x^5 + 11*x + 44 is irreducible and solvable by radicals, and (11, 44) is a primitive pair.
%t A371554 pairs = Join @@ Table[
%t A371554   Select[{m, Abs[#1 - b] #2/5} & @@@
%t A371554     Sort[SolveValues[x^2 - (6b + 5y^4)x + 25b^2 == 0 && y > 0, {x, y},
%t A371554 Integers]],
%t A371554     Max[Last /@ FactorInteger[GCD @@ #]] < 4 &&
%t A371554     AllTrue[#, IntegerQ] &&
%t A371554     IrreduciblePolynomialQ[x^5 + #1x + #2 & @@ #] &
%t A371554   ],
%t A371554   {b, 1, 1000}
%t A371554 ];
%t A371554 pairs[[All, 2]]
%Y A371554 For values of b see A371553.
%K A371554 nonn
%O A371554 1,1
%A A371554 _Ben Whitmore_, Mar 27 2024