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
Keywords
Examples
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 | +-----------------------------------------------------------+
Crossrefs
Cf. A291127.
Programs
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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: -
PARI
isok(n) = my(d=divisors(n)); polisirreducible(sum(i=1, #d, d[i]*z^(i-1))); \\ Michel Marcus, Feb 14 2025
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