

A126029


The smallest positive k such that ( sopfr(k)*tau(k) )^n = sigma(k) where sopfr is the sum of prime factors with multiplicity (A001414).


1




OFFSET

1,1


COMMENTS

35 is the only solution for n=1.
Incorrect, there are three solutions < 10^10 for n = 1: 35, 42 and 68.  Donovan Johnson, Jun 11 2013
a(3) = 14844221560107739 (conjectured) is most likely minimal but it hasn't been proved. No solutions have been found (minimal or otherwise) where the number was not squarefree.


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FORMULA



EXAMPLE

22446139 factors as: 31*67*101*107=k, sopfr(k) = sum of prime factors of k = 31+67+101+107 = 306. tau(k) = num of divisors of k = 2^4 = 16. sigma(k) = sum of divisors of k = (31+1)*(67+1)*(101+1)*(107+1) = 23970816. (306*16)^2 = 23970816. As this k turns out to be minimal, a(2)=22446139.


CROSSREFS



KEYWORD

hard,nonn,bref


AUTHOR



EXTENSIONS



STATUS

approved



