OFFSET
1,2
COMMENTS
Conjecture: a(n) > 1 except for n = 1, 5.
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).
See A385678 for a similar conjecture involving Euler's totient function.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..400
EXAMPLE
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.
MATHEMATICA
sigma[n_]:=sigma[n]=DivisorSigma[1, n];
P[n_, x_]:=P[n, x]=Sum[sigma[k]*x^(n-k), {k, 1, n}];
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]}];
tab=Append[tab, 1]; Label[aa]; Continue, {n, 1, 60}]; Print[tab]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 04 2025
STATUS
approved
