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A081705
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k-tuple abundance of abundant numbers.
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4
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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, 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, 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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| 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.
Lenstra shows that there are in fact infinitely many k-tuply abundant numbers for every k > 0.
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REFERENCES
| H. W. Lenstra, Jr., "Advanced Problems and Solutions, 6064", The American Mathematical Monthly, Vol. 84, No. 7. (Aug. - Sep., 1977), p. 580.
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FORMULA
| 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.
a(n) = number of increasing steps at the start of the aliquot chain of A005101(n).
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EXAMPLE
| 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.
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MAPLE
| A005101 := proc(n) option remember ; local a ; if n =1 then RETURN(12) ; else a := A005101(n-1)+1 ; while numtheory[sigma](a) <= 2*a do a := a+1 ; od ; RETURN(a) ; fi ; end: 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 ; end: A081705 := proc(n) aliqRis(A005101(n)) ; end: for n from 1 to 100 do printf("%d, ", A081705(n)) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 07 2007
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CROSSREFS
| Cf. A081699, A081700.
Sequence in context: A084794 A116691 A011340 * A201889 A145057 A065254
Adjacent sequences: A081702 A081703 A081704 * A081706 A081707 A081708
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KEYWORD
| nonn
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AUTHOR
| Gabriel Cunningham (gcasey(AT)mit.edu), Apr 02 2003, Dec 15 2006
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 07 2007
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