A007368 - OEIS

%I M4829 #46 Oct 21 2019 12:10:14

%S 2,1,12,24,96,72,168,240,336,360,504,576,1512,1080,1008,720,2304,3600,

%T 5376,2520,2160,1440,10416,13392,3360,4032,3024,7056,6720,2880,6480,

%U 10800,13104,5040,6048,4320,13440,5760,18720,20736,19152,22680,43680

%N Smallest k such that sigma(x) = k has exactly n solutions.

%C It's not obvious that a(n) exists for all n; I'd like to see a proof. - _David Wasserman_, Jun 07 2002

%C 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_, Jan 21 2006

%C 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

%C Index of the first occurrence of n in A054973. - _Jaroslav Krizek_, Apr 25 2009

%D 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.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe and Donovan Johnson, <a href="/A007368/b007368.txt">Table of n, a(n) for n = 0..5000</a> (terms up to a(429) from T. D. Noe)

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Kevin Ford, Sergei Konyagin, <a href="https://arxiv.org/abs/1910.08452">On two conjectures of Sierpiński concerning the arithmetic functions σ and ϕ</a>, arXiv:1910.08452 [math.NT], 2019.

%H W. Sierpiński, <a href="http://matwbn.icm.edu.pl/kstresc.php?tom=42&amp;wyd=10">Elementary Theory of Numbers</a>, Warszawa 1964, page 166.

%H R. G. Wilson, V, <a href="/A007015/a007015.pdf">Letter to N. J. A. Sloane, Jul. 1992</a>

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

%t 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}] (* _T. D. Noe_ *)

%t 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}] (* _Jean-François Alcover_, Jul 18 2017 *)

%Y Cf. A000203, A054973, A002191, A007609.

%Y Cf. A115374 (least prime p such that sigma(x)=sigma(p) has exactly n solutions).

%Y Cf. A007369, A007370, A007371, A007372 (n such that sigma(x)=k has 0, 1, 2 and 3 solutions).

%Y Cf. A184393, A184394, A201915 (smallest solution, largest solution, triangle of solutions for sigma(x)=a(n)).

%K nonn

%O 0,1

%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_

%E More terms from _David W. Wilson_