A128700 Highly abundant numbers with an odd divisor sum.

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

Offset

1,2

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.

Links

Table of n, a(n) for n=1..13.

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

Wikipedia, Highly Abundant Numbers.

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.

Example

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_.]

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, # ]] &]

Crossrefs

Cf. A002093, A000203.

Sequence in context: A076057 A364061 A133809 * A331579 A333225 A212204

Adjacent sequences: A128697 A128698 A128699 * A128701 A128702 A128703

Keyword

easy,full,nice,nonn,fini

Author

Ant King, Mar 28 2007

Status

approved