I have put together a spreadsheet in the Cloud, in Google Sheets, at this link: Prime Factorization Naming Scheme. I'm working it up as an aid in exploring the notion of constructing names for whole numbers based on their prime factorizations. The spreadsheet is visible to anyone with the link to it, but not editable. If you would like to experiment with your own ideas, just make a copy of the sheet. You should automatically get a space in Google Drive if you don't already have one.
In part, my spreadsheet was inspired by the names icarus came up with for his argam numerals (well known to members of this forum), which embody aspects of this notion. However, argam names exhibit some more artful aspects that add to their attractiveness but which make their construction more complex, and therefore harder to automate (such as with a spreadsheet). What I'm aiming for here is something along the same vein, but much more regularized.
This is also more immediately inspired by newcomer Craig's recent thread A Simple and Intuitive Name and Numeral System for Sexagesimal. Even though Craig's goal was limited to providing digits for a positional base no larger than base 60_{d}, his "datraka" naming convention does embed a similar focus on prime factorizations, with prime numbers getting more or less monosyllabic roots that are agglutinated to form the names of composite numbers. Craig's scheme is limited to giving roots only to the first three primes (2, 3, 5), but of course argam assigns roots for many, many more primes. But I was intrigued by the consonantvowel (CV) pattern Craig used, where the root for a given prime starts with a given consonant (or consonant cluster), but then the vowel indicates the power to which the prime is raised.
I've been playing around with this kind of CV pattern for a few days, but ultimately I've found it too limiting. Not enough room to cover a large number of primes in a reasonably intuitive way. So what I've opted for is something closer to argam:
 I've given each prime a monosyllabic root name with a CVC pattern, that remains stable for all of its powers. So for example 2 = bin, 3 = trin, 5 = kin, 7 = sep, 11 = lev, 13 = yon, 17 = zot, 19 = wox, etc.
 I've generally copied icarus's notion of extracting syllables from the name of chemical elements to use as names for corresponding primes. However, I've shaved off trailing vowels or added leading consonants where necessary (usually w) in order to adhere to the CVC pattern. Hence, zot rather than zote, wox rather than ax, etc.
 For raising primes to a power, I devised a common set of suffixes with a VC pattern, where the consonant is a common thematic element, but the vowel indicates the power. For example ^1 = im (usually omitted), ^2 = em, ^3 = am, ^4 = om, ^5 = um. Note that rather than ordering the vowel sounds alphabetically, I've ordered them by articulation point, from most open and fronted, to most back and rounded. My impression is that a fronted sound like /i/, with its generally higher frequency pattern, implies "smaller", while a more back sound like /u/, with its generally lower frequency pattern, implies "larger".
 To get to higher powers than five, I just introduce another thematic consonant and repeat the pattern into another senary placevalue: ^6 = ix, ^12 = ex, ^18 = ax, ^24 = ox, ^30 = ux. Combine suffixes for multiple digit powers: ix = ^6, ixim = ^7, ixem = ^8, ixam = ^9, ixom = ^10, ixum = ^11, ex = ^12, etc.
 So powers of a prime are formed by appending these suffixes, e.g.: 2 = bin, 4 = binem, 8 = binam, 16 = binom, 32 = binum, 64 = binix, 128 = binixim, 256 = binixem, etc. I've been generally placing the stress accent on the final syllable in such complexes, so that the suffixes won't all degenerate into schwa sounds.
 Composite numbers are formed by juxtaposing the roots (+ suffixes) of its prime factorization. So for example 6 = trinbin, 10 = kinbin, 12 = trinbinem, 14 = sepbin, 15 = kintrin, 18 = trinembin, 20 = kinbinem, 21 = septrin, 22 = levbin, 24 = trinbinam, 26 = yonbin, 28 = sepbinem, 30 = kintrinbin, 34 = zotbin, 35 = sepkin, 36 = trinembinem, etc. The final constant/cluster of one prime factor is immediately adjacent to the initial consonant/cluster of the next, so this makes the syllable boundaries between prime factors clear enough.
 I've opted to order the roots in a composite from largest to smallest. I found that if I did it the opposite way, every other whole number would start with bin, which gets boring; whereas, this way you get more variety up front, yet many numerals wind up rhyming at the end. One could argue that it makes sense to lead with the most significant prime. However, in principle, the order doesn't matter, because multiplication is commutative: 6 = trinbin = bintrin, 12 = trinbinem = binemtrin, etc.
 Compare argam's 2520 = kinsevoctove, vs. my equivalent sepkintrinembinam. Icarus orders from smallest to largest factor, bearing in mind that higher powers of even small primes like 2 and 3 can become larger factors. Whereas here I am ordering from largest down to smallest prime, regardless of power.
 The primorials P_{n} are: P_{1} = 2 = bin, P_{2} = 6 = trinbin, P_{3} = 30 = kintrinbin, P_{4} = 210 = sepkintrinbin, P_{5} = 2310 = levsepkintrinbin, P_{6} = 30030 = yonlevsepkintrinbin, etc.
 I've built the formulas in the spreadsheet so that once you set the name for a given prime, all its occurrences in powers and multiples automatically copy that. Likewise with the list of power suffixes (scroll to the right on the page to find columns defining those). So the spreadsheet can be used to experiment with various choices for syllables.
 After exhausting all the currentlyknown chemical elements (up to atomic number 118, oganesson → wog = 647), I went to Presidents of the United States, from George Washington → wash = 653 to Donald Trump → don = 967 (since I already had Harry S. Truman → trum = 877). Then I went to Vice Presidents of the United States (excepting those that were ever President), from Aaron Burr → bur = 971 to Michael Pence → pents = 1193. And I even dug into the Speakers of the House of Representatives of the United States, from Frederick Muhlenberger → muhl = 1201 to Paul Ryan → ryan = 1571, just 3 primes shy of 1600. (Appropriate enough, since the address of the White House is 1600 Pennsylvania Avenue.) I'd like to carry this out at least as far as P_{5} = 2310, so any suggestions for finding more sources of names would be welcome. (Hmm, maybe Supreme Court Justices? Senate majority leaders? British Prime Ministers? State names? State capitals? ...)
 This is a naming scheme only. I'm not accompanying it with any notion of how to draw numeral symbols based on their prime factorizations. I'd defer to something like UTL Numerals, which was the best scheme I've ever seen for doing that. Too bad the anonymous author only posted the images on Facebook for a brief period, until they were taken down suddenly. To my knowledge they have never resurfaced anywhere else. So all we have left is the image I put together of the first block of 210 (P_{4}) numerals. Whoever this person was, they had filled up the first catalog of 2310 (P_{5}) numerals and looked poised to go on to extend that to 30030 (P_{6}) at least. It was truly remarkable. If anyone knows who the author was, or where UTL has gone now, please let us all know!
Decimal

Dozenal

Numeral Name

1

1

un 
2

2

bin 
3

3

trin 
4

4

binem 
5

5

kin 
6

6

trinbin 
7

7

sep 
8

8

binam 
9

9

trinem 
10

A

kinbin 
11

B

lev 
12

10

trinbinem 
13

11

yon 
14

12

sepbin 
15

13

kintrin 
16

14

binom 
17

15

zot 
18

16

trinembin 
19

17

wox 
20

18

kinbinem 
21

19

septrin 
22

1A

levbin 
23

1B

fluor 
24

20

trinbinam 
25

21

kinem 
26

22

yonbin 
27

23

trinam 
28

24

sepbinem 
29

25

nev 
30

26

kintrinbin 
31

27

sod 
32

28

binum 
33

29

levtrin 
34

2A

zotbin 
35

2B

sepkin 
36

30

trinembinem 
37

31

mag 
38

32

woxbin 
39

33

yontrin 
40

34

kinbinam 
41

35

lum 
42

36

septrinbin 
43

37

sil 
44

38

levbinem 
45

39

kintrinem 
46

3A

fluorbin 
47

3B

phos 
48

40

trinbinom 
49

41

sepem 
50

42

kinembin 
51

43

zottrin 
52

44

yonbinem 
53

45

sulf 
54

46

trinambin 
55

47

levkin 
56

48

sepbinam 
57

49

woxtrin 
58

4A

nevbin 
59

4B

chlor 
60

50

kintrinbinem 
61

51

gon 
62

52

sodbin 
63

53

septrinem 
64

54

binix 
65

55

yonkin 
66

56

levtrinbin 
67

57

tash 
68

58

zotbinem 
69

59

fluortrin 
70

5A

sepkinbin 
71

5B

cal 
72

60

trinembinam 
73

61

scan 
74

62

magbin 
75

63

kinemtrin 
76

64

woxbinem 
77

65

levsep 
78

66

yontrinbin 
79

67

teit 
80

68

kinbinom 
81

69

trinom 
82

6A

lumbin 
83

6B

van 
84

70

septrinbinem 
85

71

zotkin 
86

72

silbin 
87

73

nevtrin 
88

74

levbinam 
89

75

chrom 
90

76

kintrinembin 
91

77

yonsep 
92

78

fluorbinem 
93

79

sodtrin 
94

7A

phosbin 
95

7B

woxkin 
96

80

trinbinum 
97

81

mang 
98

82

sepembin 
99

83

levtrinem 
100

84

kinembinem 
101

85

fer 
102

86

zottrinbin 
103

87

cob 
104

88

yonbinam 
105

89

sepkintrin 
106

8A

sulfbin 
107

8B

nick 
108

90

trinambinem 
109

91

cup 
110

92

levkinbin 
111

93

magtrin 
112

94

sepbinom 
113

95

zing 
114

96

woxtrinbin 
115

97

fluorkin 
116

98

nevbinem 
117

99

yontrinem 
118

9A

chlorbin 
119

9B

zotsep 
120

A0

kintrinbinam 
121

A1

levem 
122

A2

gonbin 
123

A3

lumtrin 
124

A4

sodbinem 
125

A5

kinam 
126

A6

septrinembin 
127

A7

gal 
128

A8

binixim 
129

A9

siltrin 
130

AA

yonkinbin 
131

AB

ger 
132

B0

levtrinbinem 
133

B1

woxsep 
134

B2

tashbin 
135

B3

kintrinam 
136

B4

zotbinam 
137

B5

sen 
138

B6

fluortrinbin 
139

B7

sel 
140

B8

sepkinbinem 
141

B9

phostrin 
142

BA

calbin 
143

BB

yonlev 
144

100

trinembinom 
145

101

nevkin 
146

102

scanbin 
147

103

sepemtrin 
148

104

magbinem 
149

105

brom 
150

106

kinemtrinbin 
151

107

kryp 
152

108

woxbinam 
153

109

zottrinem 
154

10A

levsepbin 
155

10B

sodkin 
156

110

yontrinbinem 
157

111

rud 
158

112

teitbin 
159

113

sulftrin 
160

114

kinbinum 
161

115

fluorsep 
162

116

trinombin 
163

117

stron 
164

118

lumbinem 
165

119

levkintrin 
166

11A

vanbin 
167

11B

yit 
168

120

septrinbinam 
169

121

yonem 
170

122

zotkinbin 
171

123

woxtrinem 
172

124

silbinem 
173

125

zirk 
174

126

nevtrinbin 
175

127

sepkinem 
176

128

levbinom 
177

129

chlortrin 
178

12A

chrombin 
179

12B

niob 
180

130

kintrinembinem 
181

131

mol 
182

132

yonsepbin 
183

133

gontrin 
184

134

fluorbinem 
185

135

magkin 
186

136

sodtrinbin 
187

137

zotlev 
188

138

phosbinem 
189

139

septrinam 
190

13A

woxkinbin 
191

13B

teck 
192

140

trinbinix 
193

141

ruth 
194

142

mangbin 
195

143

yonkintrin 
196

144

sepembinem 
197

145

rhod 
198

146

levtrinembin 
199

147

pal 
200

148

kinembinam 
201

149

tashtrin 
202

14A

ferbin 
203

14B

nevsep 
204

150

zottrinbinem 
205

151

lumkin 
206

152

cobbin 
207

153

fluortrinem 
208

154

yonbinom 
209

155

woxlev 
210

156

sepkintrinbin 