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A137825
Least number having the highest abundancy among numbers with exactly n prime factors (counted with multiplicity).
2
2, 6, 30, 60, 420, 4620, 13860, 180180, 360360, 6126120, 116396280, 2677114440, 77636318760, 155272637520, 776363187600, 24067258815600, 890488576177200, 2671465728531600, 109530094869795600, 4709794079401210800, 221360321731856907600, 11732097051788416102800
OFFSET
1,1
COMMENTS
"Least" is required in the definition, otherwise a(14) could be either 2*77636318760 or 5*77636318760, which have the same abundancy. It appears that only a(14) has this property. - T. D. Noe, Jan 24 2010
No other terms through a(800000) have the above property. - Jon E. Schoenfield, Mar 02 2019
LINKS
The Prime Glossary, Abundant Numbers
Jon E. Schoenfield, Magma program
Eric Weisstein's World of Mathematics, Abundancy
Xiaolong Wu, A New Type of Abundant Numbers, arXiv:1906.05796 [math.NT], 2019. See Table 1 p. 3.
FORMULA
a(n) = product of the first n terms of A137826. - T. D. Noe, Jan 24 2010
EXAMPLE
a(4) = 2*2*3*5 = 60 since it has four factors and its abundancy is 2.8, which is greater than that of any other number with four factors; e.g., 2*2*2*2=16, 2*2*2*3=24, 2*2*3*3=36, and 2*3*5*7=210 have abundancies 1.9375, 2.5, 2.52777..., and 2.7428571..., respectively.
PROG
(Magma) // See Schoenfield link.
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Sergio Pimentel, Feb 11 2008
EXTENSIONS
Edited by T. D. Noe, Jan 24 2010
Extended by T. D. Noe, Jan 24 2010
a(21)-a(22) from Jon E. Schoenfield, Mar 02 2019
STATUS
approved