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).

35, 22446139, 4481106818619089

Offset: 1

Author

Fred Schneider, Dec 14 2006

Keywords

• hard
• nonn
• bref

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.

a(3) <= 4481106818619089. - Donovan Johnson, Jun 10 2013

Examples

			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.
		

Links

• Mersenne Forum, Mersenne forum thread

Crossrefs

Cf. A126028, A000005, A000203, A001414, A226479, A226480.

Formula

min {k : (A001414(k)*A000005(k))^n = A000203(k)}. - R. J. Mathar, Jun 04 2013

Extensions

New name from R. J. Mathar, Jun 04 2013

a(3) from Hiroaki Yamanouchi, Sep 26 2014