Capt. Cathie Harmonic Formula One with only whole numbers output: 
Index

It may not show that we want whole numbers output but shows that it most likely can be achived. The reason we get whole number output is at least in part because all the values of C fed are squares of other number and because of the way harmonic equation one is formulated. It seems it is also a bit changed Bucky formula, could be coincidence or not that it looks like his equations. The formula below to generate C values does not give all the values for C that give whole integer values for E. The standard Fuller equation does seem to always give values that does give whole integer for E. But the equation we have used, seem to produce primes or multiplies of primes at a glance.


C= ( (2*n)*(f*(f+1))) +1

n=2 f=1 to 144 . +1 is usually +2 in buckminster fuller equations and f*f+1 is usually f*f or f**2

So C=(4*(f*(f+1)))+1

this is the computer strings used:

 >>> for x in range(1,288):
    print(x,'*',x+1,'+1 = ',fuller_nn(x,x+1,2,1),' = C ->',cathie_i(fuller_nn(x,x+1,2,1))),

and this is then the output: (which lacks, is missing 4* to the left..something else is wrong , will correct it.)
So they should look like , the first line should look like 4*(1*(2+1)) not just 1*(2+1)
   
1 * 2 +1 =  9  = C -> 36.0
2 * 3 +1 =  25  = C -> 150.0
3 * 4 +1 =  49  = C -> 392.0
4 * 5 +1 =  81  = C -> 810.0
5 * 6 +1 =  121  = C -> 1452.0
6 * 7 +1 =  169  = C -> 2366.0
7 * 8 +1 =  225  = C -> 3600.0
8 * 9 +1 =  289  = C -> 5202.0
9 * 10 +1 =  361  = C -> 7220.0
10 * 11 +1 =  441  = C -> 9702.0
11 * 12 +1 =  529  = C -> 12696.0
12 * 13 +1 =  625  = C -> 16250.0
13 * 14 +1 =  729  = C -> 20412.0
14 * 15 +1 =  841  = C -> 25230.0
15 * 16 +1 =  961  = C -> 30752.0
16 * 17 +1 =  1089  = C -> 37026.0
17 * 18 +1 =  1225  = C -> 44100.0
18 * 19 +1 =  1369  = C -> 52022.0
19 * 20 +1 =  1521  = C -> 60840.0
20 * 21 +1 =  1681  = C -> 70602.0
21 * 22 +1 =  1849  = C -> 81356.0
22 * 23 +1 =  2025  = C -> 93150.0
23 * 24 +1 =  2209  = C -> 106032.0
24 * 25 +1 =  2401  = C -> 120050.00000000001
25 * 26 +1 =  2601  = C -> 135252.0
26 * 27 +1 =  2809  = C -> 151686.0
27 * 28 +1 =  3025  = C -> 169400.0
28 * 29 +1 =  3249  = C -> 188442.0
29 * 30 +1 =  3481  = C -> 208860.0
30 * 31 +1 =  3721  = C -> 230702.0
31 * 32 +1 =  3969  = C -> 254016.0
32 * 33 +1 =  4225  = C -> 278850.0
33 * 34 +1 =  4489  = C -> 305252.0
34 * 35 +1 =  4761  = C -> 333270.0
35 * 36 +1 =  5041  = C -> 362952.0
36 * 37 +1 =  5329  = C -> 394346.0
37 * 38 +1 =  5625  = C -> 427500.0
38 * 39 +1 =  5929  = C -> 462461.99999999994
39 * 40 +1 =  6241  = C -> 499280.0
40 * 41 +1 =  6561  = C -> 538002.0
41 * 42 +1 =  6889  = C -> 578676.0
42 * 43 +1 =  7225  = C -> 621350.0
43 * 44 +1 =  7569  = C -> 666072.0
44 * 45 +1 =  7921  = C -> 712890.0
45 * 46 +1 =  8281  = C -> 761851.9999999999
46 * 47 +1 =  8649  = C -> 813005.9999999999
47 * 48 +1 =  9025  = C -> 866400.0
48 * 49 +1 =  9409  = C -> 922082.0
49 * 50 +1 =  9801  = C -> 980099.9999999999
50 * 51 +1 =  10201  = C -> 1040502.0
51 * 52 +1 =  10609  = C -> 1103336.0
52 * 53 +1 =  11025  = C -> 1168650.0
53 * 54 +1 =  11449  = C -> 1236492.0000000002
54 * 55 +1 =  11881  = C -> 1306910.0
55 * 56 +1 =  12321  = C -> 1379952.0
56 * 57 +1 =  12769  = C -> 1455666.0
57 * 58 +1 =  13225  = C -> 1534100.0
58 * 59 +1 =  13689  = C -> 1615302.0000000002
59 * 60 +1 =  14161  = C -> 1699320.0
60 * 61 +1 =  14641  = C -> 1786202.0
61 * 62 +1 =  15129  = C -> 1875995.9999999998
62 * 63 +1 =  15625  = C -> 1968750.0
63 * 64 +1 =  16129  = C -> 2064512.0
64 * 65 +1 =  16641  = C -> 2163330.0
65 * 66 +1 =  17161  = C -> 2265252.0
66 * 67 +1 =  17689  = C -> 2370326.0
67 * 68 +1 =  18225  = C -> 2478600.0
68 * 69 +1 =  18769  = C -> 2590122.0
69 * 70 +1 =  19321  = C -> 2704940.0
70 * 71 +1 =  19881  = C -> 2823102.0
71 * 72 +1 =  20449  = C -> 2944656.0
72 * 73 +1 =  21025  = C -> 3069650.0
73 * 74 +1 =  21609  = C -> 3198132.0
74 * 75 +1 =  22201  = C -> 3330150.0
75 * 76 +1 =  22801  = C -> 3465752.0
76 * 77 +1 =  23409  = C -> 3604986.0
77 * 78 +1 =  24025  = C -> 3747900.0
78 * 79 +1 =  24649  = C -> 3894541.9999999995
79 * 80 +1 =  25281  = C -> 4044960.0
80 * 81 +1 =  25921  = C -> 4199202.0
81 * 82 +1 =  26569  = C -> 4357316.0
82 * 83 +1 =  27225  = C -> 4519350.0
83 * 84 +1 =  27889  = C -> 4685352.0
84 * 85 +1 =  28561  = C -> 4855370.0
85 * 86 +1 =  29241  = C -> 5029452.0
86 * 87 +1 =  29929  = C -> 5207646.0
87 * 88 +1 =  30625  = C -> 5390000.0
88 * 89 +1 =  31329  = C -> 5576562.0
89 * 90 +1 =  32041  = C -> 5767380.0
90 * 91 +1 =  32761  = C -> 5962502.0
91 * 92 +1 =  33489  = C -> 6161976.0
92 * 93 +1 =  34225  = C -> 6365850.0
93 * 94 +1 =  34969  = C -> 6574172.0
94 * 95 +1 =  35721  = C -> 6786990.000000001
95 * 96 +1 =  36481  = C -> 7004352.0
96 * 97 +1 =  37249  = C -> 7226306.0
97 * 98 +1 =  38025  = C -> 7452900.0
98 * 99 +1 =  38809  = C -> 7684181.999999999
99 * 100 +1 =  39601  = C -> 7920200.0
100 * 101 +1 =  40401  = C -> 8161002.0
101 * 102 +1 =  41209  = C -> 8406636.0
102 * 103 +1 =  42025  = C -> 8657150.0
103 * 104 +1 =  42849  = C -> 8912592.0
104 * 105 +1 =  43681  = C -> 9173010.0
105 * 106 +1 =  44521  = C -> 9438452.0
106 * 107 +1 =  45369  = C -> 9708966.0
107 * 108 +1 =  46225  = C -> 9984600.0
108 * 109 +1 =  47089  = C -> 10265402.0
109 * 110 +1 =  47961  = C -> 10551420.0
110 * 111 +1 =  48841  = C -> 10842702.000000002
111 * 112 +1 =  49729  = C -> 11139296.0
112 * 113 +1 =  50625  = C -> 11441250.0
113 * 114 +1 =  51529  = C -> 11748612.000000002
114 * 115 +1 =  52441  = C -> 12061430.0
115 * 116 +1 =  53361  = C -> 12379752.0
116 * 117 +1 =  54289  = C -> 12703626.0
117 * 118 +1 =  55225  = C -> 13033100.0
118 * 119 +1 =  56169  = C -> 13368221.999999998
119 * 120 +1 =  57121  = C -> 13709040.000000002
120 * 121 +1 =  58081  = C -> 14055602.0
121 * 122 +1 =  59049  = C -> 14407955.999999998
122 * 123 +1 =  60025  = C -> 14766149.999999998
123 * 124 +1 =  61009  = C -> 15130232.0
124 * 125 +1 =  62001  = C -> 15500249.999999998
125 * 126 +1 =  63001  = C -> 15876252.0
126 * 127 +1 =  64009  = C -> 16258285.999999998
127 * 128 +1 =  65025  = C -> 16646400.0
128 * 129 +1 =  66049  = C -> 17040642.0
129 * 130 +1 =  67081  = C -> 17441060.0
130 * 131 +1 =  68121  = C -> 17847702.0
131 * 132 +1 =  69169  = C -> 18260616.0
132 * 133 +1 =  70225  = C -> 18679850.0
133 * 134 +1 =  71289  = C -> 19105452.0
134 * 135 +1 =  72361  = C -> 19537470.0
135 * 136 +1 =  73441  = C -> 19975952.0
136 * 137 +1 =  74529  = C -> 20420946.0
137 * 138 +1 =  75625  = C -> 20872500.0
138 * 139 +1 =  76729  = C -> 21330662.0
139 * 140 +1 =  77841  = C -> 21795480.0
140 * 141 +1 =  78961  = C -> 22267002.0
141 * 142 +1 =  80089  = C -> 22745276.0
142 * 143 +1 =  81225  = C -> 23230350.0
143 * 144 +1 =  82369  = C -> 23722272.0
144 * 145 +1 =  83521  = C -> 24221090.0
145 * 146 +1 =  84681  = C -> 24726852.0
146 * 147 +1 =  85849  = C -> 25239606.0
147 * 148 +1 =  87025  = C -> 25759400.0
148 * 149 +1 =  88209  = C -> 26286282.000000004
149 * 150 +1 =  89401  = C -> 26820300.0
150 * 151 +1 =  90601  = C -> 27361502.0
151 * 152 +1 =  91809  = C -> 27909936.0
152 * 153 +1 =  93025  = C -> 28465650.0
153 * 154 +1 =  94249  = C -> 29028692.0
154 * 155 +1 =  95481  = C -> 29599110.0
155 * 156 +1 =  96721  = C -> 30176952.0
156 * 157 +1 =  97969  = C -> 30762266.0
157 * 158 +1 =  99225  = C -> 31355100.0
158 * 159 +1 =  100489  = C -> 31955502.0
159 * 160 +1 =  101761  = C -> 32563520.0
160 * 161 +1 =  103041  = C -> 33179202.0
161 * 162 +1 =  104329  = C -> 33802596.0
162 * 163 +1 =  105625  = C -> 34433750.0
163 * 164 +1 =  106929  = C -> 35072712.0
164 * 165 +1 =  108241  = C -> 35719530.0
165 * 166 +1 =  109561  = C -> 36374252.0
166 * 167 +1 =  110889  = C -> 37036926.0
167 * 168 +1 =  112225  = C -> 37707600.0
168 * 169 +1 =  113569  = C -> 38386321.99999999
169 * 170 +1 =  114921  = C -> 39073140.0
170 * 171 +1 =  116281  = C -> 39768102.0
171 * 172 +1 =  117649  = C -> 40471256.0
172 * 173 +1 =  119025  = C -> 41182650.0
173 * 174 +1 =  120409  = C -> 41902332.0
174 * 175 +1 =  121801  = C -> 42630350.0
175 * 176 +1 =  123201  = C -> 43366752.0
176 * 177 +1 =  124609  = C -> 44111585.99999999
177 * 178 +1 =  126025  = C -> 44864900.0
178 * 179 +1 =  127449  = C -> 45626742.0
179 * 180 +1 =  128881  = C -> 46397160.0
180 * 181 +1 =  130321  = C -> 47176202.0
181 * 182 +1 =  131769  = C -> 47963916.0
182 * 183 +1 =  133225  = C -> 48760350.0
183 * 184 +1 =  134689  = C -> 49565552.0
184 * 185 +1 =  136161  = C -> 50379570.0
185 * 186 +1 =  137641  = C -> 51202452.0
186 * 187 +1 =  139129  = C -> 52034246.0
187 * 188 +1 =  140625  = C -> 52875000.0
188 * 189 +1 =  142129  = C -> 53724762.0
189 * 190 +1 =  143641  = C -> 54583580.0
190 * 191 +1 =  145161  = C -> 55451502.0
191 * 192 +1 =  146689  = C -> 56328576.0
192 * 193 +1 =  148225  = C -> 57214850.0
193 * 194 +1 =  149769  = C -> 58110372.0
194 * 195 +1 =  151321  = C -> 59015190.00000001
195 * 196 +1 =  152881  = C -> 59929352.00000001
196 * 197 +1 =  154449  = C -> 60852906.00000001
197 * 198 +1 =  156025  = C -> 61785900.0
198 * 199 +1 =  157609  = C -> 62728382.0
199 * 200 +1 =  159201  = C -> 63680400.0
200 * 201 +1 =  160801  = C -> 64642002.0
201 * 202 +1 =  162409  = C -> 65613236.0
202 * 203 +1 =  164025  = C -> 66594150.0
203 * 204 +1 =  165649  = C -> 67584792.0
204 * 205 +1 =  167281  = C -> 68585210.0
205 * 206 +1 =  168921  = C -> 69595452.0
206 * 207 +1 =  170569  = C -> 70615566.0
207 * 208 +1 =  172225  = C -> 71645600.0
208 * 209 +1 =  173889  = C -> 72685602.0
209 * 210 +1 =  175561  = C -> 73735620.0
210 * 211 +1 =  177241  = C -> 74795702.0
211 * 212 +1 =  178929  = C -> 75865896.0
212 * 213 +1 =  180625  = C -> 76946250.0
213 * 214 +1 =  182329  = C -> 78036812.0
214 * 215 +1 =  184041  = C -> 79137630.0
215 * 216 +1 =  185761  = C -> 80248752.0
216 * 217 +1 =  187489  = C -> 81370226.0
217 * 218 +1 =  189225  = C -> 82502100.0
218 * 219 +1 =  190969  = C -> 83644422.0
219 * 220 +1 =  192721  = C -> 84797240.0
220 * 221 +1 =  194481  = C -> 85960602.0
221 * 222 +1 =  196249  = C -> 87134556.00000001
222 * 223 +1 =  198025  = C -> 88319150.00000001
223 * 224 +1 =  199809  = C -> 89514432.0
224 * 225 +1 =  201601  = C -> 90720450.0
225 * 226 +1 =  203401  = C -> 91937252.0
226 * 227 +1 =  205209  = C -> 93164886.0
227 * 228 +1 =  207025  = C -> 94403400.0
228 * 229 +1 =  208849  = C -> 95652842.0
229 * 230 +1 =  210681  = C -> 96913260.00000001
230 * 231 +1 =  212521  = C -> 98184702.0
231 * 232 +1 =  214369  = C -> 99467216.0
232 * 233 +1 =  216225  = C -> 100760850.0
233 * 234 +1 =  218089  = C -> 102065652.0
234 * 235 +1 =  219961  = C -> 103381670.0
235 * 236 +1 =  221841  = C -> 104708952.0
236 * 237 +1 =  223729  = C -> 106047546.0
237 * 238 +1 =  225625  = C -> 107397500.0
238 * 239 +1 =  227529  = C -> 108758862.0
239 * 240 +1 =  229441  = C -> 110131679.99999999
240 * 241 +1 =  231361  = C -> 111516002.0
241 * 242 +1 =  233289  = C -> 112911876.00000001
242 * 243 +1 =  235225  = C -> 114319350.0
243 * 244 +1 =  237169  = C -> 115738472.0
244 * 245 +1 =  239121  = C -> 117169290.00000001
245 * 246 +1 =  241081  = C -> 118611851.99999999
246 * 247 +1 =  243049  = C -> 120066206.0
247 * 248 +1 =  245025  = C -> 121532400.0
248 * 249 +1 =  247009  = C -> 123010482.00000001
249 * 250 +1 =  249001  = C -> 124500499.99999999
250 * 251 +1 =  251001  = C -> 126002501.99999999
251 * 252 +1 =  253009  = C -> 127516535.99999999
252 * 253 +1 =  255025  = C -> 129042650.0
253 * 254 +1 =  257049  = C -> 130580892.0
254 * 255 +1 =  259081  = C -> 132131309.99999999
255 * 256 +1 =  261121  = C -> 133693952.0
256 * 257 +1 =  263169  = C -> 135268866.0
257 * 258 +1 =  265225  = C -> 136856100.0
258 * 259 +1 =  267289  = C -> 138455702.0
259 * 260 +1 =  269361  = C -> 140067720.0
260 * 261 +1 =  271441  = C -> 141692202.0
261 * 262 +1 =  273529  = C -> 143329196.0
262 * 263 +1 =  275625  = C -> 144978750.0
263 * 264 +1 =  277729  = C -> 146640912.0
264 * 265 +1 =  279841  = C -> 148315730.0
265 * 266 +1 =  281961  = C -> 150003252.0
266 * 267 +1 =  284089  = C -> 151703526.0
267 * 268 +1 =  286225  = C -> 153416600.0
268 * 269 +1 =  288369  = C -> 155142522.0
269 * 270 +1 =  290521  = C -> 156881340.0
270 * 271 +1 =  292681  = C -> 158633102.0
271 * 272 +1 =  294849  = C -> 160397856.0
272 * 273 +1 =  297025  = C -> 162175650.0
273 * 274 +1 =  299209  = C -> 163966532.0
274 * 275 +1 =  301401  = C -> 165770550.0
275 * 276 +1 =  303601  = C -> 167587752.0
276 * 277 +1 =  305809  = C -> 169418186.0
277 * 278 +1 =  308025  = C -> 171261900.0
278 * 279 +1 =  310249  = C -> 173118942.0
279 * 280 +1 =  312481  = C -> 174989360.0
280 * 281 +1 =  314721  = C -> 176873202.0
281 * 282 +1 =  316969  = C -> 178770516.0
282 * 283 +1 =  319225  = C -> 180681350.0
283 * 284 +1 =  321489  = C -> 182605752.0
284 * 285 +1 =  323761  = C -> 184543770.0
285 * 286 +1 =  326041  = C -> 186495452.0
286 * 287 +1 =  328329  = C -> 188460846.0
287 * 288 +1 =  330625  = C -> 190440000.0