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A386827 Least prime n < p < 2*n*(n-1) such that the polynomial Sum_{k=1..n} x^(n-k)/k is irreducible modulo p, or 1 if such a prime p does not exist.

%I A386827 #17 Aug 11 2025 23:51:32
%S A386827 1,3,7,13,7,11,83,11,43,103,41,29,89,67,43,23,41,67,131,269,47,151,43,
%T A386827 149,191,127,29,113,263,173,61,463,223,67,61,127,103,97,47,271,89,59,
%U A386827 337,281,157,541,269,277,73,337,463,379,223,1481,827,797,397,101,337,431
%N A386827 Least prime n < p < 2*n*(n-1) such that the polynomial Sum_{k=1..n} x^(n-k)/k is irreducible modulo p, or 1 if such a prime p does not exist.
%C A386827 Conjecture: a(n) > 1 for all n > 1. In other words, for each n = 2,3,... there is a prime p with n < p < 2*n*(n-1) such that the polynomial Sum_{k=1..n} x^(n-k)/k is irreducible modulo p.
%H A386827 Zhi-Wei Sun, <a href="/A386827/b386827.txt">Table of n, a(n) for n = 1..400</a>
%H A386827 Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/498716">On the polynomial x^{n-1}+x^{n-2}/2+...+1/n</a>, Question 498716 in MathOverflow, August 5, 2025.
%e A386827 a(7) = 83 since 83 = 2*7*(7-1) - 1 is the least prime p > 7 such that the polynomial x^6 + x^5/2 + x^4/3 + x^3/4 + x^2/5 + x/6 + 1/7 is irreducible modulo p.
%t A386827 P[n_, x_]:=P[n, x]=Sum[x^(n-k)/k, {k, 1, n}];
%t A386827 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(n-1)-1]}];
%t A386827 tab=Append[tab,1]; Label[aa]; Continue, {n,1,60}];Print[tab]
%o A386827 (PARI) a(n) = forprime(p=n+1, 2*n*(n-1)-1, if (polisirreducible(Mod(sum(k=1, n, x^(n-k)/k), p)), return(p))); 1; \\ _Michel Marcus_, Aug 05 2025
%Y A386827 Cf. A000040, A385658, A385676, A385678, A386828, A386850.
%K A386827 nonn
%O A386827 1,2
%A A386827 _Zhi-Wei Sun_, Aug 04 2025