A381097 Consider the polynomial P(m,z) = Sum_{i=1..k} d(i)*z^(i-1) where d(1), d(2), ..., d(k) are the k divisors of m. The sequence lists the numbers m such that P(m,z) is irreducible.

2, 3, 4, 5, 7, 9, 11, 12, 13, 16, 17, 19, 23, 24, 25, 29, 30, 31, 36, 37, 40, 41, 43, 45, 47, 48, 49, 53, 56, 59, 60, 61, 63, 64, 67, 70, 71, 72, 73, 79, 80, 81, 83, 84, 89, 90, 96, 97, 101, 103, 105, 107, 108, 109, 112, 113, 120, 121, 126, 127, 131, 132, 135

Offset

1,1

Comments

The squares>1 and the prime numbers are in the sequence.

Links

Table of n, a(n) for n=1..63.

Example

The prime numbers q are in the sequence because P(q,z) = qz + 1.
6 is not in the sequence because P(6,z)=(2z+1)*(3z^2+1).
The following table gives the irreducible polynomials.
+-----------------------------------------------------------+
|  m |                 P(m,z)                               |
+-----------------------------------------------------------+
|  4 | 1 + 2z + 4z^2                                        |
+-----------------------------------------------------------+
|  9 | 1 + 3z + 9z^2                                        |
+-----------------------------------------------------------+
| 12 | 1 + 2z + 3z^2 + 4z^3 + 6z^4 + 12z^5                  |
+-----------------------------------------------------------+
| 16 | 1 + 2z + 4z^2 + 8z^3 + 16z^4                         |
+----------------------------+------------------------------+
| 24 | 1 + 2z + 3z^2 + 4z^3 + 6z^4 + 8z^5 + 12z^6 + 24z^7   |
+-----------------------------------------------------------+
| 25 | 1 + 5z + 25z^2                                       |
+-----------------------------------------------------------+
| 30 | 1 + 2z + 3z^2 + 5z^3 + 6z^4 + 10z^5 + 15z^6 + 30z^7  |
+-----------------------------------------------------------+

Maple

with(numtheory):
for n from 1 to 135 do :
  it:=0:d:=divisors(n):P:=add(op(i, d)*x^(i-1), i=1..nops(d)):
   y:=fsolve(P, x, complex):z:=evalf({%}):k:=nops(z):
    if irreduc(P) then printf(`%d, `, n):else fi:
od:

Prog

(PARI) isok(n) = my(d=divisors(n)); polisirreducible(sum(i=1, #d, d[i]*z^(i-1))); \\ Michel Marcus, Feb 14 2025

Crossrefs

Cf. A291127.

Sequence in context: A286972 A210994 A331593 * A319237 A331230 A381540

Adjacent sequences: A381094 A381095 A381096 * A381098 A381099 A381100

Keyword

nonn,new

Author

Michel Lagneau, Feb 14 2025

Status

approved