theory - Complexity of bin packing with defined function of bin weight -

i'm struggling following problem:

given n integers, place them m bins, total sum in bins minimized. trick once numbers placed in bin, total weight/cost/sum of bin computed in non-standard way:

weight_of_bin = sigma - k * x sigma sum of integers in bin k number of integers in bin x number of prime divisors integers located in bin have in common.

in other words, grouping numbers have many prime divisors in common, , placing different quantities of numbers in different bins, can achieve "savings" in total sum.

i use bin-packing formulation because suspect problem nphard have trouble finding proof. not number theory person , confused fact weight of bin depends on items in bin.

are there hardness results type of problem?

p.s. know numbers integers. there no explicit limit on largest integer involved in problem.

thanks pointers can give.

this not complete answer, gives things think about.

first, way of clarification: know prime divisors of integers? finding prime divisors of integers in input problem difficult enough is. factorization isn't known np-complete, isn't known in p. if don't know factorization of inputs, might enough make problem "hard".

in general, expect problem @ least hard bin packing. simple argument show is possible none of integers given have common prime divisors (for example, if given set of distinct primes). in case, problem reduces standard bin packing since weight of bin standard weight. if have guarantee how many common divisors there may be, may possibly better, not in general.

there variant of bin packing, called vm packing (based on idea of packing virtual machines based on memory requirements) objects allowed share space (such shared virtual memory pages). objective function, subtract term based on "shared" prime divisors reminds me of that. in case of vm packing, problem np-hard. if sharing has nice hierarchy, approximation algorithms exist, still approximations.