A128700 Highly abundant numbers with an odd divisor sum.

1, 2, 4, 8, 16, 18, 36, 72, 144, 288, 1800, 3600, 7200

Offset: 1

Author

Ant King, Mar 28 2007

Keywords

• easy
• full
• nice
• nonn
• fini

Comments

Alaoglu and Erdős showed that 7200 is the largest highly abundant number with all the exponents of its prime factors occurring to powers greater than unity. It follows that the sequence of highly abundant numbers with an odd divisor sum is finite and is bounded above by 7200. Accordingly, this is the complete sequence of such integers.

Examples

			The fifth highly abundant number with an odd divisor sum is 16. Hence a(5)=16. [Corrected by _N. J. A. Sloane_, Jan 11 2024 at the suggestion of _Harvey P.Dale_.]
		

Links

• L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
• Wikipedia, Highly Abundant Numbers.

Crossrefs

Cf. A002093, A000203.

Programs

Mathematica

hadata1=FoldList[Max,1,Table[DivisorSigma[1,n],{n,2,7200}]]; data1=Flatten[Position[hadata1,#,1,1]&/@Union[hadata1]];Select[data1,OddQ[DivisorSigma[1,# ]] &]

Formula

The highly abundant numbers are those integers for which sigma(n) > sigma(m) for all m < n (A002093). This sequence contains those elements of A002093 that have an odd divisor sum.