I am interested in Primorial number system (primoradic, see stub OEIS). In that system, you can define "primorial fractions" as fractions which can be writen as $$ \frac{{a_1 }}{2} + \frac{{a_2 }}{6} + \frac{{a_3 }}{{30}} + \cdots + \frac{{a_n }}{{P(p_n )}} $$ with $a_1$ in $\left\{ {0,1} \right\}$, $a_2$ in $\left\{ {0,1,2} \right\}$, $a_3$ in $\left\{ {0,1,2,3,4} \right\}$, $\ldots$ , $a_n$ in $\left\{ {0,1,2,\ldots,p_n -1} \right\}$ and P($p_n$)=$\prod_{i=1}^n p_i$. For example, $\frac{7}{10}=\frac{21}{30}=\frac{15+5+1}{30}=\frac{15}{30}+\frac{5}{30}+\frac{1}{30}=\frac{1}{2}+\frac{1}{6}+\frac{1}{30}$ is a primorial fraction. $\frac{1}{4}$=$\frac{1}{2}\times\frac{1}{2}$ is not a primorial fraction even if $\frac{1}{2}$ IS a primorial fraction: $\frac 14=\sum_{k=2}^{\infty}\frac{\left(\frac{p_k-1}{2}\right)}{P(p_k)}$("primorial development" of $\frac 14$)
I propose an extension for the notations adopted in stub OEIS concerning Primorial number system : for example, 2+$\frac{1}{2}$+$\frac{1}{6}$+$\frac{1}{30}$=2.7=[2,1:1:1]. So that [2,1:1:1]<e. We can then write $\frac 14$=[0,0:1:2:3:5:6:8:9:.......:$\frac{p_k-1}{2}$:...]. My question is then : what is the primorial development of e ? https://oeis.org/wiki/Primorial_numeral_system P.S. :I find, as a beginning, e=[2,1:1:1:3:9:3:0:1:1:16:25:...]. What about Log2=[0,1:1:0:5:...]?Log6=[1,1:1:3:...]?Log(30)? Log(210)?...$\frac{2310}{Log(2310)}$?
See Primorial expansion of e.
For $\log(2)$ the first $100$ coefficients $\{a_1, a_2, a_3 ..., a_{100}\}$ are as follows:
$\{1, 1, 0, 5, 6, 2, 3, 10, 17, 24, 14, 12, 8, 41, 38, 45, 49, 0, 59, \ 17, 37, 63, 49, 58, 3, 96, 39, 82, 50, 45, 117, 10, 45, 131, 52, 108, \ 143, 14, 93, 43, 87, 8, 115, 114, 63, 97, 0, 148, 57, 53, 116, 161, \ 186, 245, 45, 65, 233, 266, 71, 208, 277, 120, 207, 196, 162, 257, \ 72, 178, 35, 81, 171, 56, 331, 157, 343, 289, 312, 87, 372, 402, 354, \ 169, 317, 165, 311, 62, 209, 176, 112, 362, 161, 256, 287, 51, 341, \ 221, 473, 505, 189, 135\}$
Here's the Mathematica function I defined to derive the initial term followed by the first $K$ coefficients $\{a_1, a_2, a_3 ..., a_K\}$ of the Primorial expansion of $x$:
primorialFractionExpansion[x_,K_]:=Block[{k=1,den=2,sum=Floor[x],y,outList={}}, While[k<=K, y=Min[Floor[(x-sum)den],Prime[k]-1]; outList=AppendTo[outList,y]; sum+=y/den; k++; den*=Prime[k]]; {Floor[x],outList} ]
Here are a few more primorial expansions for various values of $x$ in the format $x:\{a_0=\lfloor x\rfloor,\{a_1,a_2,a_3,...,a_{50}\}\}$:
$\log(6):\ \{1,\{1,1,3,5,2,12,9,2,9,9,26,16,22,27,46,8,29,46,65,18,60,25,36,13,54,51,6,47,42,8,68,3,61,132,29,0,142,24,12,167,60,127,91,102,148,188,164,140,131,2\}\}$
$\log(30):\ \{3,\{0,2,2,0,2,9,16,5,5,9,1,1,11,18,6,13,48,1,61,17,41,62,70,35,78,64,78,23,15,20,104,88,47,46,43,43,109,108,153,85,91,67,39,163,49,180,146,119,103,214\}\}$
$\log (210):\ \{5,\{0,2,0,2,9,10,10,16,10,6,17,18,34,42,18,38,38,59,24,15,49,29,28,61,69,54,98,62,52,86,117,25,97,4,108,120,41,161,138,126,25,6,42,183,147,28,196,25,45,198\}\}$
$\log (2310):\ \{7,\{1,1,2,2,4,12,7,7,5,17,7,13,14,14,22,34,8,56,10,6,8,60,17,53,82,39,39,83,12,43,41,117,16,119,8,132,145,16,103,83,78,88,175,173,118,166,177,27,56,76\}\}$
$\frac{2310}{\log(2310)}:\ \{298,\{0,1,2,4,10,3,11,5,17,22,5,11,17,42,12,21,32,20,14,66,28,72,46,74,89,15,29,101,88,53,22,28,9,88,65,63,30,45,65,159,152,57,107,47,99,84,110,97,6,227\}\}$