Re: problems with sum functions/ factoring the factorial

• To: mathgroup at smc.vnet.net
• Subject: [mg65852] Re: problems with sum functions/ factoring the factorial
• From: Roger Bagula <rlbagulatftn at yahoo.com>
• Date: Tue, 18 Apr 2006 06:56:49 -0400 (EDT)
• References: <e1sns1\$86r\$1@smc.vnet.net> <e1vhef\$a94\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```This cf[n] is in the OEIS already - see A049614
Roger Bagula wrote:
> Another  inferior definition of the primorial function exists at:
> http://mathworld.wolfram.com/Primorial.html
> There are in fact 396 sequences in OEIS that mention the name:
> http://www.research.att.com/~njas/sequences/?q=primorial&language=english&go=Search
> The one I'm refering to is:
> http://www.research.att.com/~njas/sequences/?q=A034386&sort=0&fmt=0&language=english&go=Search
> Also see:
> http://www.research.att.com/~njas/sequences/?q=A117683&sort=0&fmt=0&language=english&go=Search
> Roger
> Roger Bagula wrote:
>
>
>>I have no problem constructing the Primorial or Compositorial functions
>>that factor the factorial as:
>>cf[n]*p[n]=n!
>>But trying to get the sine and cosine constructed functions to plot
>>seems to be a problem here:
>>Clear[f, g, cf, p, CeS, CeC, PeS, PeC]
>>f[n_] := If[PrimeQ[n] == True, 1, n]
>>cf[0] = 1;
>>cf[n_Integer?Positive] := cf[n] = f[n]*cf[n - 1]
>>g[n_] := If[PrimeQ[n] == True, n, 1]
>>p[0] = 1;
>>p[n_Integer?Positive] := p[n] = g[n]*p[n - 1]
>>Ce = 1 + Sum[1/cf[n], {n, 1, 1000}];
>>N[%, 100]
>>Pe = 1 + Sum[1/p[n], {n, 1, 1000}];
>>N[%, 100]
>>CeS[x_] := 1 + NSum[(-1)^n*p[2*n + 1]*x^(2*n + 1)/(2*n + 1)!, {n, 1, 100}];
>>CeC[x_] := 1 + NSum[(-1)^n*p[2*n]*x^(2*n)/(2*n)!, {n, 1, 100}];
>>ParametricPlot[{CeC[x], CeS[x]}, {x, 0, 2*Pi}]
>>PeS[x_] := 1 + NSum[(-1)^n*cf[2*n + 1]*x^(2*n + 1)/(2*n + 1)!, {n, 1, 100}];
>>PeC[x_] := 1 + NSum[(-1)^n*cf[2*n]*x^(2*n)/(2*n)!, {n, 1, 100}];
>>ParametricPlot[{PeC[x], PeS[x]}, {x, 0, 2*Pi}]
>>
>>Alernative functions:
>>CeS1[x_] := 1 + NSum[(-1)^n*x^(2*n + 1)/cf[2*n + 1], {n, 1, 100}];
>>CeC1[x_] := 1 + NSum[(-1)^n*x^(2*n)/cf[2*n], {n, 1, 100}];
>>ParametricPlot[{CeC[x], CeS[x]}, {x, 0, 2*Pi}]
>>PeS1[x_] := 1 + NSum[(-1)^n*x^(2*n + 1)/p[2*n + 1], {n, 1, 100}];
>>PeC1[x_] := 1 + NSum[(-1)^n*x^(2*n)/p[2*n], {n, 1, 100}];
>>ParametricPlot[{PeC1[x], PeS1[x]}, {x, 0, 2*Pi}]
>>
>>In addition this function seem to come up with the wrong sign:
>>Pe[x_] := 1 + NSum[cf[n]*x^n/n!, {n, 1, 100}];
>>Plot[Pe[x], {x, 0, 5}]
>>
>>Or alternatively:
>>Pe1[x_] := 1 + NSum[x^n/p[n], {n, 1, 100}];
>>Plot[Pe1[x], {x, 0, 5}]
>>
>>Ce*Pe~ 5*E (low)
>>Roger
>>
>
>

```

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