A123523 Smallest odd number k such that sigma(x) = k has exactly n solutions.

1, 31, 347529, 10773399, 4104665019, 77253471477, 28732655133, 35492068813383, 108695634368139, 461396894573979, 68452476460273269, 2529134502772059, 99414440839732473

Offset: 1

Author

T. D. Noe, Oct 02 2006

Keywords

• more
• nonn

Comments

Note that sigma(x) is odd iff x is in A028982 (numbers of the form m^2 or 2m^2 for m > 0).

a(14) > 10^18. a(15) = 175792216832685999. a(16) > 10^18. - Donovan Johnson, Jun 09 2011

From David A. Corneth, Apr 27 2019: (Start)

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:

a(14) <= 2489145199534927711323, for n = 16..27, a(n) <= 30520233337797869211, 1292387730916522149, 3939513268555279291149, 1066776514086397590567, 7538497634436073695117, 1629700928685734429889, 7217246969893966760937, 136456488459785229549035859, 396763033391372299743, 2215694819757447795607659, 500318185106520469975923, 5916133590898752361467873 respectively.

All these listed upper bounds are divisors of 12302819034343122006137404371659222028537. No more divisors of this number are an upper bound for any n.

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)

Examples

			For n = 3, sigma(x) = 347529 has exactly three solutions x = 164836, 203522, 239121.
		

Links

• Max Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions, Journal of Integer Sequences 19 (2016), Article 16.5.2
• David A. Corneth, PARI program, partially written by Max Alekseyev

Crossrefs

Cf. A007368, A300869.

Different from A123524.

Programs

Mathematica

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]]}]

Extensions

a(8) from Martin Fuller, Oct 07 2006

a(9)-a(10) from Donovan Johnson, Dec 09 2008

a(11)-a(13) from Donovan Johnson, Jun 09 2011