

A007368


Smallest k such that sigma(x) = k has exactly n solutions.
(Formerly M4829)


24



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
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OFFSET

0,1


COMMENTS

It's not obvious that a(n) exists for all n; I'd like to see a proof.  David Wasserman, Jun 07 2002
Note that k1 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, Jan 21 2006
According to Sierpiński, H. J. Kanold proved that there is a k such that sigma(x)=k has n or more solutions. Sierpiński states that Erdős 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, Oct 18 2006


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).


LINKS

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].


EXAMPLE

a(10) = 504; {204, 220, 224, 246, 284, 286, 334, 415, 451, 503} is the set of x such that sigma(x) = 504.


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, firstMissing1}] (* T. D. Noe *)
terms = 100; cnt = DivisorSigma[1, Range[terms^3]] // Tally // Sort; a[0] = 2; a[n_] := SelectFirst[cnt, #[[2]] == n&][[1]]; Table[a[n], {n, 0, terms  1}] (* JeanFrançois Alcover, Jul 18 2017 *)


CROSSREFS

Cf. A115374 (least prime p such that sigma(x)=sigma(p) has exactly n solutions).
Cf. A184393, A184394, A201915 (smallest solution, largest solution, triangle of solutions for sigma(x)=a(n)).


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



