File:Dickman prob factor size.svg
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Size of this PNG preview of this SVG file: 800 × 512 pixels. Other resolutions: 320 × 205 pixels | 640 × 409 pixels | 1,024 × 655 pixels | 1,280 × 818 pixels | 2,560 × 1,637 pixels | 1,076 × 688 pixels.
Original file (SVG file, nominally 1,076 × 688 pixels, file size: 41 KB)
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Summary[edit]
| Description |
English: Plot of probability that the largest and second largest prime factor of x is less than x^a
With p(u) = Dickman function, plot(p(1/a), a = 0 to 1) Based on solving the delay differential equations |
| Date | |
| Source | Own work |
| Author | SethT |
Based on a similar graphic from Knuth 4.5.4 Fig 11, p. 362
Source Code[edit]
| Mathematica source code |
|---|
uMax := 100;
sol = NDSolve[
{u*p'[u] + p[u - 1] == 0,
p[u /; u <= 1] == 1,
u*q'[u] + q[u - 1] - p[u - 1] == 0,
q[u /; u <= 1] == 1},
{p, q},
{u, 0, uMax},
WorkingPrecision -> 40, PrecisionGoal -> 20];
NP[u_] := Piecewise[{{p[u] /. First[sol], u <= uMax}}, 0]
(* Use derivative for very small a via interpolatated large u *)
NQ[u_] :=
Piecewise[{{q[u] /. First[sol],
u <= uMax}}, (1/u)*(q[uMax] /. First[sol])*uMax]
F2[a_] := NP[1/a]
G2[a_] := NQ[1/a]
Plot[{F2[a], G2[a]}, {a, 0, 1}, PlotRange -> All,
Ticks -> {{#, x^#} & /@ Range[0, 1, 0.1], Automatic},
BaseStyle -> {FontWeight -> Bold, FontSize -> 12},
PlotStyle -> Blue, GridLines -> All, PlotRangePadding -> 0.004]
LogPlot[{NP[u]}, {u, 0, 15}]
Plot[{NQ[u]}, {u, 0, uMax}, PlotRange -> Full] |
Licensing[edit]
I, the copyright holder of this work, hereby publish it under the following license:
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- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 08:38, 22 April 2019 | | 1,076 × 688 (41 KB) | SethT (talk | contribs) | Slightly better display of function at y=0 and y=1 |
| 08:35, 22 April 2019 |  | 1,076 × 693 (45 KB) | SethT (talk | contribs) | {{Information |description ={{en|1=Plot of probability that the largest and second largest prime factor of x is less than x^a With p(u) = Dickman function, plot(p(1/a), a = 0 to 1) Based on solving the delay differential equations :<math>u\rho'(u) + \rho(u-1) = 0\,</math> :<math>u\tau'(u) + \tau(u-1) - \rho(u-1) = 0</math> with initial conditions <math>\rho(u) = \tau(u) = 1</math> for 0 ≤ ''u'' ≤ 1.}} |date =2019-04-21 |source ={{own}} |author =... |
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