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Question Number 5777 by FilupSmith last updated on 27/May/16
∃x∈Z+:x=p1p2...pn∧∀p≠x,p∈Z+x=(factorsofp1)(fact.p2)...(fact.pn)Howcouldyouformallywritethenumberofwaysyoucanwritexintermsoftheproductofnintegers?
Commented by FilupSmith last updated on 27/May/16
ω(x)=numberofdistinctprimefactorse.g.12=23×31ω(12)=2x=∏ω(x)t=1ptet(productofprimefactors)let℧(x)=totalnumberofprimefactors℧(x)=∑ω(x)t=1et∴combinationsC(x)=(℧(x))!C(x)=(∑ω(x)t=1et)!????
ifcombinationsoftwoproductsx=pqx=(p1p2...pm)(q1q2...qn)=(m+n)!=(℧(p)+℧(q))!forx=abc...nC(x)=(℧(a)+℧(b)+...+℧(n))!ithinkthisisthecase
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