File:Dickman prob factor size.svg
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Summary
| 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
| 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
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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 | Slightly better display of function at y=0 and y=1 |
| 08:35, 22 April 2019 |  | 1,076 × 693 (45 KB) | SethT | {{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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