A090052 Group-abundant numbers: n such that the number of groups of order n (A000001) exceeds n.

32, 48, 64, 96, 128, 144, 160, 192, 256, 288, 320, 384, 432, 448, 480, 512, 576, 640, 648, 672, 704, 720, 768, 800, 832, 864, 896, 960, 1024, 1088, 1152, 1216, 1248, 1280, 1296, 1344, 1408, 1440, 1458, 1536, 1600, 1664, 1728, 1792, 1920, 1944, 2016, 2048, 2112, 2160, 2176, 2187, 2240, 2304, 2400, 2432, 2496, 2560, 2592, 2688, 2816, 2880, 2916, 2944

Offset: 1

Author

J. H. Conway, Jan 21 2004

Keywords

• nonn

Comments

It seems fairly certain that 1 is the only group-perfect number and that almost all numbers are group-deficient. However, all that is known at present is that all squarefree numbers except 1 are group-deficient.

Examples

			32 is in the sequence because A000001(32) = 51 > 32, 48 is in the sequence because A000001(48) = 52 > 48 and since the exact number of groups of order 2048 that have exponent-2 class 2 is 1774274116992170 then 2048 is in the sequence because A000001(2048) > 1774274116992170 > 2048. - _Muniru A Asiru_, Nov 26 2017
		

Links

• Alex Meiburg, Table of n, a(n) for n = 1..178
• J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.

Crossrefs

Cf. A000001.

Programs

GAP

A090052 := Filtered([1..2015], n -> NumberSmallGroups(n) > n); # Muniru A Asiru, Nov 30 2017

Extensions

1944, 2016, and 2048 added by Eric M. Schmidt, Aug 02 2012

a(49)-a(52) from Muniru A Asiru, Nov 26 2017

a(53)-a(178) from Alex Meiburg, Dec 30 2017, partially using https://github.com/olexandr-konovalov/gnu/