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.

1, 2, 3, 2, 1, 5, 11, 29, 2, 47, 5, 31, 13, 379, 37, 251, 23, 29, 67, 97, 41, 131, 11, 173, 41, 139, 79, 103, 281, 19, 7, 53, 71, 281, 131, 19, 3, 43, 149, 23, 347, 47, 29, 107, 107, 47, 823, 47, 311, 547, 67, 419, 263, 379, 349, 23, 227, 349, 19, 113

Offset: 1

Author

Zhi-Wei Sun, Aug 04 2025

Keywords

• nonn

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.

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..400

Crossrefs

Cf. A000040, A000203, A385658, A385678.

Programs

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]

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