A137825 Least number having the highest abundancy among numbers with exactly n prime factors (counted with multiplicity).

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

Jon E. Schoenfield, Table of n, a(n) for n = 1..373

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

Cf. A005101, A017665, A017666, A137826.

Sequence in context: A360680 A369685 A127517 * A008341 A211889 A174276

Adjacent sequences: A137822 A137823 A137824 * A137826 A137827 A137828

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