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.
1, 3, 7, 13, 7, 11, 83, 11, 43, 103, 41, 29, 89, 67, 43, 23, 41, 67, 131, 269, 47, 151, 43, 149, 191, 127, 29, 113, 263, 173, 61, 463, 223, 67, 61, 127, 103, 97, 47, 271, 89, 59, 337, 281, 157, 541, 269, 277, 73, 337, 463, 379, 223, 1481, 827, 797, 397, 101, 337, 431
Offset: 1
Keywords
Examples
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.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..400
- Zhi-Wei Sun, On the polynomial x^{n-1}+x^{n-2}/2+...+1/n, Question 498716 in MathOverflow, August 5, 2025.
Programs
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Mathematica
P[n_, x_]:=P[n, x]=Sum[x^(n-k)/k, {k, 1, n}]; 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]}]; tab=Append[tab,1]; Label[aa]; Continue, {n,1,60}];Print[tab] -
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
Comments