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A386828 Least prime n < p <= 2*n^2 + 1 such that the polynomial Sum_{k=1..n} x^(n-k)/k^2 is irreducible modulo p, or 1 if such a prime p does not exist.

%I A386828 #9 Aug 11 2025 23:25:12
%S A386828 1,3,19,13,7,17,19,13,17,13,17,31,139,151,19,181,113,157,79,89,89,71,
%T A386828 37,31,197,31,199,149,83,37,127,59,647,89,47,47,157,197,97,79,601,59,
%U A386828 79,67,71,487,223,577,359,83,269,269,251,461,229,67,1777,859,1091,701
%N A386828 Least prime n < p <= 2*n^2 + 1 such that the polynomial Sum_{k=1..n} x^(n-k)/k^2 is irreducible modulo p, or 1 if such a prime p does not exist.
%C A386828 Conjecture: a(n) > 1 for all n > 1.
%C A386828 We also have similar conjectures for Sum_{k=1..n} x^(n-k)/k^s with other values of s.
%H A386828 Zhi-Wei Sun, <a href="/A386828/b386828.txt">Table of n, a(n) for n = 1..300</a>
%e A386828 a(3) = 19 since 19 = 2*3^2 + 1 is the least prime p > 3 such that the polynomial x^2 + x/2 + 1/3 is irreducible modulo p.
%t A386828 P[n_, x_]:=P[n, x]=Sum[x^(n-k)/k^2, {k, 1, n}];
%t A386828 tab={};Do[Do[If[IrreduciblePolynomialQ[P[n, x], Modulus->Prime[k]]==True, tab=Append[tab,Prime[k]]; Goto[aa]], {k, PrimePi[n]+1, PrimePi[2n^2+1]}]; tab=Append[tab,1]; Label[aa]; Continue, {n,1,60}];Print[tab]
%o A386828 (PARI) a(n) = forprime(p=n+1, 2*n^2+1, if (polisirreducible(Mod(sum(k=1, n, x^(n-k)/k^2), p)), return(p))); 1; \\ _Michel Marcus_, Aug 05 2025
%Y A386828 Cf. A000040, A386827.
%K A386828 nonn
%O A386828 1,2
%A A386828 _Zhi-Wei Sun_, Aug 05 2025