Let $\ n\ge 2\ $ be an integer and $\ d_1,\cdots,d_k\ $ its positive divisors in increasing order. Define $$f(x)=\sum_{j=0}^{k-1} d_{k-j}x^j$$ In other words, we form a polynomial of the positive divisors of $\ n\ $. Example $$f_{12}=x^5 + 2x^4 + 3x^3 + 4x^2 + 6x + 12$$
Can we classify the composite numbers $n$ such that $f_n$ is irreducible over $\mathbb Z[x]$ ?
For primes $\ n\ $ , $\ f_n\ $ is clearly irreducible because its degree is $\ 1\ $. The composite values $\ n\ $ upto $\ 200\ $ giving an irreducible polynomial are :
4 9 12 16 24 25 30 36 40 45 48 49 56 60 63 64 70 72 80 81 84 90 96 105 108 112 120 121 126 132 135 140 144 150 154 160 165 168 169 175 176 180 182 189 192 195 198 200
Any ideas or references ?