A128699 Highly abundant numbers that are not superabundant, i.e., the complement of A004394 w.r.t. A002093.

3, 8, 10, 16, 18, 20, 30, 42, 72, 84, 90, 96, 108, 144, 168, 210, 216, 288, 300, 336, 420, 480, 504, 540, 600, 630, 660, 960, 1008, 1080, 1200, 1440, 1560, 1620, 1800, 1920, 1980, 2100, 2160, 2340, 2400, 2880, 3024, 3120, 3240, 3360, 3600, 3780, 3960, 4200

Offset: 1

Author

Ant King, Mar 28 2007

Keywords

• nonn

Comments

In 1944, Alaoglu and Erdős conjectured that this sequence was infinite and this was proved to be true by Nicolas in 1969.

Examples

			The sequence of highly abundant numbers begins 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20 and the sequence of superabundant numbers begins 1, 2, 4, 6, 12, 24. Because 10 is the third number which is in the first sequence but not in the second, it follows that a(3)=10.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
• Jean-Louis Nicolas, Ordre maximal d'un élément du groupe Sn des permutations et "highly composite numbers" Bull. Soc. Math. France 97 (1969), pp. 129-191.
• Eric Weisstein's World of Mathematics, Superabundant Number.
• Wikipedia, Highly Abundant Numbers.

Crossrefs

Cf. A000203, A002093, A004394, A128700, A128701, A128702.

Programs

Mathematica

habdata1=FoldList[Max,1,Table[DivisorSigma[1,n],{n,2,10000}]]; data1=Flatten[Position[habdata1,#,1,1]&/@Union[habdata1]];sabdata2=FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10000}]]; data2=Flatten[Position[sabdata2,#,1,1]&/@Union[sabdata2]];sabdata2=FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10000}]]; Complement[data1,data2]

Formula

The highly abundant numbers are those integers for which sigma(n) > sigma(m) for all m < n (A002093) and the superabundant numbers are those integers for which sigma(n)/n > sigma(m)/m for all m < n (A004394).