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A385676 Least prime p <= 2*n^2 - n + 1 such that the polynomial Sum_{k=1..n} sigma(k) * x^(n-k) is irreducible modulo p, or 1 if such p does not exist, where sigma is given by A000203.

%I A385676 #18 Aug 11 2025 05:07:58
%S A385676 1,2,3,2,1,5,11,29,2,47,5,31,13,379,37,251,23,29,67,97,41,131,11,173,
%T A385676 41,139,79,103,281,19,7,53,71,281,131,19,3,43,149,23,347,47,29,107,
%U A385676 107,47,823,47,311,547,67,419,263,379,349,23,227,349,19,113
%N A385676 Least prime p <= 2*n^2 - n + 1 such that the polynomial Sum_{k=1..n} sigma(k) * x^(n-k) is irreducible modulo p, or 1 if such p does not exist, where sigma is given by A000203.
%C A385676 Conjecture: a(n) > 1 except for n = 1, 5.
%C A385676 Note that Sum_{k=1..5} sigma(k) * x^(5-k) = x^4 + 3*x^3 + 4*x^2 + 7*x + 6 = (x + 2)*(x^3 + x^2 + 2*x + 3).
%C A385676 See A385678 for a similar conjecture involving Euler's totient function.
%H A385676 Zhi-Wei Sun, <a href="/A385676/b385676.txt">Table of n, a(n) for n = 1..400</a>
%e A385676 a(14) = 379 since 379 = 2*14^2 - 14 + 1 is the least prime p such that Sum_{k=1..14} sigma(k) * x^(14-k) is irreducible modulo p.
%t A385676 sigma[n_]:=sigma[n]=DivisorSigma[1,n];
%t A385676 P[n_, x_]:=P[n, x]=Sum[sigma[k]*x^(n-k), {k, 1, n}];
%t A385676 tab={};Do[Do[If[IrreduciblePolynomialQ[P[n, x], Modulus->Prime[k]]==True, tab=Append[tab,Prime[k]]; Goto[aa]], {k, 1, PrimePi[2n^2-n+1]}];
%t A385676 tab=Append[tab,1]; Label[aa]; Continue, {n, 1, 60}];Print[tab]
%o A385676 (PARI) a(n) = forprime(p=2, 2*n^2 - n + 1, if (polisirreducible(Mod(sum(k=1, n, sigma(k)*x^(n-k)), p)), return(p))); 1; \\ _Michel Marcus_, Aug 04 2025
%Y A385676 Cf. A000040, A000203, A385658, A385678.
%K A385676 nonn
%O A385676 1,2
%A A385676 _Zhi-Wei Sun_, Aug 04 2025