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

1,2

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)

Links

Table of n, a(n) for n=1..13.

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

Example

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

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]]}]

Crossrefs

Cf. A007368, A300869.

Different from A123524.

Sequence in context: A059113 A057839 A123524 * A079595 A051155 A342118

Adjacent sequences: A123520 A123521 A123522 * A123524 A123525 A123526

Keyword

more,nonn

Author

T. D. Noe, Oct 02 2006

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

Status

approved