
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
We call a positive integer  $n$   $\textit{sixish}$  if  $n=p(p+6)$ , where  $p$  and  $p+6$  are prime numbers. For example,  $187=11\cdot17$  is sixish, but  $475=19\cdot25$  is not sixish. Define a function  $f$  on the positive integers such that  $f(n)$  is the sum of the squares of the positive divisors of  $n$ . For example,  $f(10)=1^2+2^2+5^2+10^2=130$ .

(a) Find, with proof, an irreducible polynomial function  $g(x)$  with integer coefficients such that  $f(n)=g(n)$  for all sixish  $n$ . ("Irreducible" means that  $g(x)$  cannot be factored as the product of two polynomials of smaller degree with integer coefficients.)

(b) We call a positive integer  $n$   $\textit{pseudo-sixish}$  if  $n$  is not sixish but nonetheless  $f(n)=g(n)$ , where  $g(n)$  is the polynomial function that you found in part (a). Find, with proof, all pseudo-sixish positive integers.

Present the answer in LaTex format: \boxed{Your answer}
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