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A007368
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Smallest k such that sigma(x) = k has exactly n solutions.
(Formerly M4829)
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14
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2, 1, 12, 24, 96, 72, 168, 240, 336, 360, 504, 576, 1512, 1080, 1008, 720, 2304, 3600, 5376, 2520, 2160, 1440, 10416, 13392, 3360, 4032, 3024, 7056, 6720, 2880, 6480, 10800, 13104, 5040, 6048, 4320, 13440, 5760, 18720, 20736, 19152, 22680, 43680
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| It's not obvious that a(n) exists for all n; I'd like to see a proof. - David Wasserman (wasserma(AT)spawar.navy.mil), Jun 07 2002
Note that k-1 is frequently prime. See A115374 for the least prime. For each n, it appears that there are an infinite number of k such that sigma(x)=k has exactly n solutions. - T. D. Noe (noe(AT)sspectra.com), Jan 21 2006
According to Sierpinski, H. J. Kanold proved that there is a k such that sigma(x)=k has n or more solutions. Sierpinski states that Erdos proved that if, for some k, sigma(x)=k has exactly n solutions, then there are an infinite number of such k. - T. D. Noe (noe(AT)sspectra.com), Oct 18 2006
Index of the first occurrence of n in A054973. - Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Apr 25 2009
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 0..5000 (terms up to a(429) from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Sierpi\'{n}ski, Elementary Theory of Numbers, Warszawa 1964, page 166.
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EXAMPLE
| a(10) = 504; {204, 220, 224, 246, 284, 286, 334, 415, 451, 503} is the set of x such that sigma(x) = 504.
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MATHEMATICA
| Needs["Statistics`DataManipulation`"]; s=DivisorSigma[1, Range[10^5]]; f=Frequencies[s]; fs=Sort[f]; tfs=Transpose[fs][[1]]; utfs=Union[tfs]; firstMissing=First[Complement[Range[Last[utfs]], utfs]]; pos=1; Table[While[tfs[[pos]]<n, pos++ ]; fs[[pos, 2]], {n, firstMissing-1}] (Noe)
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CROSSREFS
| Cf. A000203, A054973, A002191, A007609.
Cf. A115374 (least prime p such that sigma(x)=sigma(p) has exactly n solutions).
Cf. A007369, A007370, A007371, A007372 (n such that sigma(x)=k has 0, 1, 2 and 3 solutions).
Sequence in context: A045873 A110060 A061081 * A054677 A012929 A013161
Adjacent sequences: A007365 A007366 A007367 * A007369 A007370 A007371
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein, Robert G. Wilson v (rgwv(AT)rgwv.com)
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EXTENSIONS
| More terms from David W. Wilson (davidwwilson(AT) comcast.net).
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