|
%I #49 May 06 2019 01:58:01
%S 1,31,347529,10773399,4104665019,77253471477,28732655133,
%T 35492068813383,108695634368139,461396894573979,68452476460273269,
%U 2529134502772059,99414440839732473
%N Smallest odd number k such that sigma(x) = k has exactly n solutions.
%C Note that sigma(x) is odd iff x is in A028982 (numbers of the form m^2 or 2m^2 for m > 0).
%C a(14) > 10^18. a(15) = 175792216832685999. a(16) > 10^18. - _Donovan Johnson_, Jun 09 2011
%C From _David A. Corneth_, Apr 27 2019: (Start)
%C The least common divisor of the first 13 terms is k = 63540409508528099686942221. Checking the divisors of k to see if they give an upper bound for some a(n) gives these upper bounds:
%C a(14) <= 2489145199534927711323, for n = 16..27, a(n) <= 30520233337797869211, 1292387730916522149, 3939513268555279291149, 1066776514086397590567, 7538497634436073695117, 1629700928685734429889, 7217246969893966760937, 136456488459785229549035859, 396763033391372299743, 2215694819757447795607659, 500318185106520469975923, 5916133590898752361467873 respectively.
%C All these listed upper bounds are divisors of 12302819034343122006137404371659222028537. No more divisors of this number are an upper bound for any n.
%C This method doesn't give a stronger lower bound except that it tells us that a new upper bound for some term is no divisor of k. (End)
%H Max Alekseyev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Alekseyev/alek5.html">Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions</a>, Journal of Integer Sequences 19 (2016), Article 16.5.2
%H David A. Corneth, <a href="/A123523/a123523_2.gp.txt">PARI program, partially written by Max Alekseyev</a>
%e For n = 3, sigma(x) = 347529 has exactly three solutions x = 164836, 203522, 239121.
%t Needs["Statistics`DataManipulation`"]; nn=10^6; t1=DivisorSigma[1,Range[nn]^2]; t2=DivisorSigma[1,2*Range[nn/Sqrt[2]]^2]; t=Join[t1,t2]; {u,v}=Transpose[Sort[Frequencies[t]]]; Table[p=Position[u,i,1,1][[1,1]]; v[[p]], {i,Length[Union[u]]}]
%Y Cf. A007368, A300869.
%Y Different from A123524.
%K more,nonn
%O 1,2
%A _T. D. Noe_, Oct 02 2006
%E a(8) from _Martin Fuller_, Oct 07 2006
%E a(9)-a(10) from _Donovan Johnson_, Dec 09 2008
%E a(11)-a(13) from _Donovan Johnson_, Jun 09 2011
| 