A128700 - OEIS

%I #11 Jan 11 2024 20:08:33

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

%N Highly abundant numbers with an odd divisor sum.

%C 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.

%H L. Alaoglu and P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1944-03.pdf">On highly composite and similar numbers</a>, Trans. Amer. Math. Soc., 56 (1944), 448-469.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Highly_abundant_number">Highly Abundant Numbers</a>.

%F 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.

%e 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_.]

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

%Y Cf. A002093, A000203.

%K easy,full,nice,nonn,fini

%O 1,2

%A _Ant King_, Mar 28 2007