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A081705 k-tuple abundance of abundant numbers.

%I A081705 #16 Oct 11 2017 06:56:44
%S A081705 1,1,1,2,7,1,1,6,1,5,1,2,4,1,1,3,1,3,1,2,2,1,7,1,1,1,6,8,5,3,31,2,1,
%T A081705 30,1,1,1,28,5,2,14,4,1,1,3,1,2,1,14,4,1,29,4,1,28,7,4,5,3,1,11,2,1,6,
%U A081705 3,12,1,11,1,1,6,5,27,18,1,1,17,1,2,3,3,1,1,14,4,4,13,1,1,12,2,3,10,1,5,1,4
%N A081705 k-tuple abundance of abundant numbers.
%C A081705 Note that only increasing steps at the beginning of an aliquot chain count toward k-tuple abundance. I wonder if there are an infinite number of k-tuply abundant numbers for all k? Another interesting question - are there any numbers that are completely abundant (that is, numbers whose aliquot chain increases forever). Though there are several numbers whose aliquot chains aren't yet fully determined, all the ones I've checked have had a finite k-tuple abundance.
%C A081705 Lenstra shows that there are in fact infinitely many k-tuply abundant numbers for every k > 0.
%H A081705 Paolo P. Lava, <a href="/A081705/b081705.txt">Table of n, a(n) for n = 1..275 (first 97 terms from Gabriel Cunningham)</a>
%H A081705 H. W. Lenstra, Jr., <a href="http://www.jstor.org/stable/2320042">Advanced Problems and Solutions, 6064</a>, The American Mathematical Monthly, Vol. 84, No. 7. (Aug. - Sep., 1977), p. 580.
%F A081705 a(n) = 0 if n is not abundant, otherwise 1 + (a(sigma(n)-n)) Note, however, that non-abundant numbers are excluded from this sequence.
%F A081705 a(n) = number of increasing steps at the start of the aliquot chain of A005101(n).
%e A081705 a(4)=2 because the 4th abundant number is 24 which has aliquot sequence 24->36->55->17->1, which has two increasing steps at the beginning.
%p A081705 aliqRis := proc(n) local r,a,an ; r := 0 ; a := n; while true do an := numtheory[sigma](a)-a ; if an > a then r := r+1 ; a := an ; else RETURN(r) ; fi ; od ;
%p A081705 end proc:
%p A081705 A081705 := proc(n)
%p A081705     aliqRis(A005101(n)) ;
%p A081705 end proc:
%p A081705 seq(A081705(n),n=1..100) ; # _R. J. Mathar_, Mar 07 2007
%Y A081705 Cf. A081699, A081700.
%K A081705 nonn
%O A081705 1,4
%A A081705 Gabriel Cunningham (gcasey(AT)mit.edu), Apr 02 2003, Dec 15 2006
%E A081705 More terms from _R. J. Mathar_, Mar 07 2007