A137825 - OEIS

%I #29 Mar 31 2024 12:02:36

%S 2,6,30,60,420,4620,13860,180180,360360,6126120,116396280,2677114440,

%T 77636318760,155272637520,776363187600,24067258815600,890488576177200,

%U 2671465728531600,109530094869795600,4709794079401210800,221360321731856907600,11732097051788416102800

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

%C "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

%C No other terms through a(800000) have the above property. - _Jon E. Schoenfield_, Mar 02 2019

%H Jon E. Schoenfield, <a href="/A137825/b137825.txt">Table of n, a(n) for n = 1..373</a>

%H The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=AbundantNumber">Abundant Numbers</a>

%H Jon E. Schoenfield, <a href="/A137825/a137825.txt">Magma program</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Abundancy.html">Abundancy</a>

%H Xiaolong Wu, <a href="https://arxiv.org/abs/1906.05796">A New Type of Abundant Numbers</a>, arXiv:1906.05796 [math.NT], 2019. See Table 1 p. 3.

%F a(n) = product of the first n terms of A137826. - _T. D. Noe_, Jan 24 2010

%e 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.

%o (Magma) // See Schoenfield link.

%Y Cf. A005101, A017665, A017666, A137826.

%K easy,nonn

%O 1,1

%A _Sergio Pimentel_, Feb 11 2008

%E Edited by _T. D. Noe_, Jan 24 2010

%E Extended by _T. D. Noe_, Jan 24 2010

%E a(21)-a(22) from _Jon E. Schoenfield_, Mar 02 2019