Conference rusure::math

1066.0. "Primes in [2 .. 10000000]" by FOO::BHAVNANI (SYS$UNWIND - laid back VMS) Fri Apr 21 1989 14:49

	I've computed the primes between 2 and 10,000,000.  The largest of
	the 664579 primes found is 9999991.  The program took 1 min 37 secs
	of clock time to execute on an 8840 with 47 reasonably active users.
	A summary of the distribution follows the formfeed.

	If anyone's interested, send me mail and I'll copy the output to
	your DECnet dir.  The file is 13200 blocks long.  Each page contains
	60 lines with 8 primes per line.  There is blank line after every
	5 lines.  When printed at 8 pages per sheet on an LPS40 printer,
	the output was about an inch thick and took 2+ hours to print.

	I hope to optimize my data structures so that I can compute primes
	upto 80,000,000.

	/ravi


1000	primes in [2 .. 7919]
2000	primes in [2 .. 17389]
3000	primes in [2 .. 27449]
4000	primes in [2 .. 37813]
5000	primes in [2 .. 48611]
6000	primes in [2 .. 59359]
7000	primes in [2 .. 70657]
8000	primes in [2 .. 81799]
9000	primes in [2 .. 93179]
10000	primes in [2 .. 104729]
11000	primes in [2 .. 116447]
12000	primes in [2 .. 128189]
13000	primes in [2 .. 139901]
14000	primes in [2 .. 151703]
15000	primes in [2 .. 163841]
16000	primes in [2 .. 176081]
17000	primes in [2 .. 187963]
18000	primes in [2 .. 200183]
19000	primes in [2 .. 212369]
20000	primes in [2 .. 224737]
21000	primes in [2 .. 237203]
22000	primes in [2 .. 249439]
23000	primes in [2 .. 262139]
24000	primes in [2 .. 274529]
25000	primes in [2 .. 287117]
26000	primes in [2 .. 300023]
27000	primes in [2 .. 312583]
28000	primes in [2 .. 324949]
29000	primes in [2 .. 337541]
30000	primes in [2 .. 350377]
31000	primes in [2 .. 363269]
32000	primes in [2 .. 376127]
33000	primes in [2 .. 389171]
34000	primes in [2 .. 401987]
35000	primes in [2 .. 414977]
36000	primes in [2 .. 427991]
37000	primes in [2 .. 440723]
38000	primes in [2 .. 453889]
39000	primes in [2 .. 467213]
40000	primes in [2 .. 479909]
41000	primes in [2 .. 493127]
42000	primes in [2 .. 506131]
43000	primes in [2 .. 519227]
44000	primes in [2 .. 532333]
45000	primes in [2 .. 545747]
46000	primes in [2 .. 559081]
47000	primes in [2 .. 572311]
48000	primes in [2 .. 585493]
49000	primes in [2 .. 598687]
50000	primes in [2 .. 611953]
51000	primes in [2 .. 625187]
52000	primes in [2 .. 638977]
53000	primes in [2 .. 652429]
54000	primes in [2 .. 665659]
55000	primes in [2 .. 679277]
56000	primes in [2 .. 692543]
57000	primes in [2 .. 706019]
58000	primes in [2 .. 719639]
59000	primes in [2 .. 732923]
60000	primes in [2 .. 746773]
61000	primes in [2 .. 760267]
62000	primes in [2 .. 773603]
63000	primes in [2 .. 787207]
64000	primes in [2 .. 800573]
65000	primes in [2 .. 814279]
66000	primes in [2 .. 827719]
67000	primes in [2 .. 841459]
68000	primes in [2 .. 855359]
69000	primes in [2 .. 868771]
70000	primes in [2 .. 882377]
71000	primes in [2 .. 896009]
72000	primes in [2 .. 909683]
73000	primes in [2 .. 923591]
74000	primes in [2 .. 937379]
75000	primes in [2 .. 951161]
76000	primes in [2 .. 965113]
77000	primes in [2 .. 978947]
78000	primes in [2 .. 993107]
79000	primes in [2 .. 1006721]
80000	primes in [2 .. 1020379]
81000	primes in [2 .. 1034221]
82000	primes in [2 .. 1048129]
83000	primes in [2 .. 1062511]
84000	primes in [2 .. 1076143]
85000	primes in [2 .. 1090373]
86000	primes in [2 .. 1103923]
87000	primes in [2 .. 1117579]
88000	primes in [2 .. 1131617]
89000	primes in [2 .. 1145689]
90000	primes in [2 .. 1159523]
91000	primes in [2 .. 1173301]
92000	primes in [2 .. 1187003]
93000	primes in [2 .. 1200949]
94000	primes in [2 .. 1215133]
95000	primes in [2 .. 1229269]
96000	primes in [2 .. 1243709]
97000	primes in [2 .. 1257517]
98000	primes in [2 .. 1271293]
99000	primes in [2 .. 1285517]
100000	primes in [2 .. 1299709]
101000	primes in [2 .. 1313839]
102000	primes in [2 .. 1327901]
103000	primes in [2 .. 1342051]
104000	primes in [2 .. 1356227]
105000	primes in [2 .. 1370459]
106000	primes in [2 .. 1384631]
107000	primes in [2 .. 1398367]
108000	primes in [2 .. 1412641]
109000	primes in [2 .. 1427039]
110000	primes in [2 .. 1441049]
111000	primes in [2 .. 1455089]
112000	primes in [2 .. 1469393]
113000	primes in [2 .. 1483451]
114000	primes in [2 .. 1497541]
115000	primes in [2 .. 1511539]
116000	primes in [2 .. 1525921]
117000	primes in [2 .. 1540499]
118000	primes in [2 .. 1554881]
119000	primes in [2 .. 1569173]
120000	primes in [2 .. 1583539]
121000	primes in [2 .. 1598011]
122000	primes in [2 .. 1612133]
123000	primes in [2 .. 1626479]
124000	primes in [2 .. 1640833]
125000	primes in [2 .. 1655131]
126000	primes in [2 .. 1669313]
127000	primes in [2 .. 1683673]
128000	primes in [2 .. 1698077]
129000	primes in [2 .. 1712353]
130000	primes in [2 .. 1726943]
131000	primes in [2 .. 1741409]
132000	primes in [2 .. 1755563]
133000	primes in [2 .. 1770437]
134000	primes in [2 .. 1784633]
135000	primes in [2 .. 1798967]
136000	primes in [2 .. 1813351]
137000	primes in [2 .. 1827659]
138000	primes in [2 .. 1842349]
139000	primes in [2 .. 1856297]
140000	primes in [2 .. 1870667]
141000	primes in [2 .. 1885201]
142000	primes in [2 .. 1899481]
143000	primes in [2 .. 1913983]
144000	primes in [2 .. 1928257]
145000	primes in [2 .. 1942747]
146000	primes in [2 .. 1957129]
147000	primes in [2 .. 1972177]
148000	primes in [2 .. 1986757]
149000	primes in [2 .. 2000963]
150000	primes in [2 .. 2015177]
151000	primes in [2 .. 2029717]
152000	primes in [2 .. 2044127]
153000	primes in [2 .. 2058871]
154000	primes in [2 .. 2073349]
155000	primes in [2 .. 2088133]
156000	primes in [2 .. 2102717]
157000	primes in [2 .. 2117131]
158000	primes in [2 .. 2131813]
159000	primes in [2 .. 2146231]
160000	primes in [2 .. 2160553]
161000	primes in [2 .. 2175451]
162000	primes in [2 .. 2190031]
163000	primes in [2 .. 2204827]
164000	primes in [2 .. 2219641]
165000	primes in [2 .. 2234129]
166000	primes in [2 .. 2248723]
167000	primes in [2 .. 2263381]
168000	primes in [2 .. 2277553]
169000	primes in [2 .. 2292469]
170000	primes in [2 .. 2307229]
171000	primes in [2 .. 2322143]
172000	primes in [2 .. 2336861]
173000	primes in [2 .. 2351821]
174000	primes in [2 .. 2366297]
175000	primes in [2 .. 2381147]
176000	primes in [2 .. 2395867]
177000	primes in [2 .. 2410621]
178000	primes in [2 .. 2425229]
179000	primes in [2 .. 2440219]
180000	primes in [2 .. 2454587]
181000	primes in [2 .. 2469413]
182000	primes in [2 .. 2484331]
183000	primes in [2 .. 2499023]
184000	primes in [2 .. 2513591]
185000	primes in [2 .. 2528411]
186000	primes in [2 .. 2543221]
187000	primes in [2 .. 2558009]
188000	primes in [2 .. 2572649]
189000	primes in [2 .. 2587007]
190000	primes in [2 .. 2601857]
191000	primes in [2 .. 2616703]
192000	primes in [2 .. 2631529]
193000	primes in [2 .. 2646089]
194000	primes in [2 .. 2660753]
195000	primes in [2 .. 2675909]
196000	primes in [2 .. 2690557]
197000	primes in [2 .. 2705243]
198000	primes in [2 .. 2719631]
199000	primes in [2 .. 2735129]
200000	primes in [2 .. 2750159]
201000	primes in [2 .. 2764873]
202000	primes in [2 .. 2779771]
203000	primes in [2 .. 2794723]
204000	primes in [2 .. 2809309]
205000	primes in [2 .. 2823589]
206000	primes in [2 .. 2838169]
207000	primes in [2 .. 2853187]
208000	primes in [2 .. 2868043]
209000	primes in [2 .. 2883371]
210000	primes in [2 .. 2898527]
211000	primes in [2 .. 2913259]
212000	primes in [2 .. 2928151]
213000	primes in [2 .. 2943257]
214000	primes in [2 .. 2958383]
215000	primes in [2 .. 2973059]
216000	primes in [2 .. 2987843]
217000	primes in [2 .. 3002743]
218000	primes in [2 .. 3017689]
219000	primes in [2 .. 3032831]
220000	primes in [2 .. 3047767]
221000	primes in [2 .. 3062989]
222000	primes in [2 .. 3077687]
223000	primes in [2 .. 3092371]
224000	primes in [2 .. 3107633]
225000	primes in [2 .. 3122321]
226000	primes in [2 .. 3137447]
227000	primes in [2 .. 3152341]
228000	primes in [2 .. 3167141]
229000	primes in [2 .. 3182341]
230000	primes in [2 .. 3196933]
231000	primes in [2 .. 3211441]
232000	primes in [2 .. 3226423]
233000	primes in [2 .. 3241201]
234000	primes in [2 .. 3256423]
235000	primes in [2 .. 3271277]
236000	primes in [2 .. 3286331]
237000	primes in [2 .. 3301591]
238000	primes in [2 .. 3316427]
239000	primes in [2 .. 3331423]
240000	primes in [2 .. 3346601]
241000	primes in [2 .. 3362159]
242000	primes in [2 .. 3376991]
243000	primes in [2 .. 3391831]
244000	primes in [2 .. 3406801]
245000	primes in [2 .. 3421751]
246000	primes in [2 .. 3436571]
247000	primes in [2 .. 3451229]
248000	primes in [2 .. 3466829]
249000	primes in [2 .. 3482333]
250000	primes in [2 .. 3497861]
251000	primes in [2 .. 3512869]
252000	primes in [2 .. 3528403]
253000	primes in [2 .. 3543791]
254000	primes in [2 .. 3558913]
255000	primes in [2 .. 3573877]
256000	primes in [2 .. 3588941]
257000	primes in [2 .. 3604121]
258000	primes in [2 .. 3618497]
259000	primes in [2 .. 3633347]
260000	primes in [2 .. 3648923]
261000	primes in [2 .. 3664351]
262000	primes in [2 .. 3678931]
263000	primes in [2 .. 3694217]
264000	primes in [2 .. 3709073]
265000	primes in [2 .. 3724223]
266000	primes in [2 .. 3738937]
267000	primes in [2 .. 3754087]
268000	primes in [2 .. 3769763]
269000	primes in [2 .. 3784943]
270000	primes in [2 .. 3800201]
271000	primes in [2 .. 3815209]
272000	primes in [2 .. 3830003]
273000	primes in [2 .. 3845273]
274000	primes in [2 .. 3860347]
275000	primes in [2 .. 3875827]
276000	primes in [2 .. 3890969]
277000	primes in [2 .. 3906319]
278000	primes in [2 .. 3921217]
279000	primes in [2 .. 3936301]
280000	primes in [2 .. 3951161]
281000	primes in [2 .. 3967043]
282000	primes in [2 .. 3981883]
283000	primes in [2 .. 3997859]
284000	primes in [2 .. 4012847]
285000	primes in [2 .. 4027913]
286000	primes in [2 .. 4042729]
287000	primes in [2 .. 4057967]
288000	primes in [2 .. 4073233]
289000	primes in [2 .. 4088237]
290000	primes in [2 .. 4103629]
291000	primes in [2 .. 4119053]
292000	primes in [2 .. 4134629]
293000	primes in [2 .. 4149889]
294000	primes in [2 .. 4164637]
295000	primes in [2 .. 4180097]
296000	primes in [2 .. 4195057]
297000	primes in [2 .. 4210253]
298000	primes in [2 .. 4225457]
299000	primes in [2 .. 4241099]
300000	primes in [2 .. 4256233]
301000	primes in [2 .. 4271581]
302000	primes in [2 .. 4287253]
303000	primes in [2 .. 4302631]
304000	primes in [2 .. 4317823]
305000	primes in [2 .. 4333097]
306000	primes in [2 .. 4348691]
307000	primes in [2 .. 4364111]
308000	primes in [2 .. 4379467]
309000	primes in [2 .. 4394783]
310000	primes in [2 .. 4410317]
311000	primes in [2 .. 4425107]
312000	primes in [2 .. 4440041]
313000	primes in [2 .. 4455043]
314000	primes in [2 .. 4470287]
315000	primes in [2 .. 4485479]
316000	primes in [2 .. 4500731]
317000	primes in [2 .. 4516231]
318000	primes in [2 .. 4531811]
319000	primes in [2 .. 4547239]
320000	primes in [2 .. 4562693]
321000	primes in [2 .. 4577927]
322000	primes in [2 .. 4593089]
323000	primes in [2 .. 4608493]
324000	primes in [2 .. 4623793]
325000	primes in [2 .. 4639267]
326000	primes in [2 .. 4654697]
327000	primes in [2 .. 4670207]
328000	primes in [2 .. 4685257]
329000	primes in [2 .. 4700473]
330000	primes in [2 .. 4716053]
331000	primes in [2 .. 4731271]
332000	primes in [2 .. 4746551]
333000	primes in [2 .. 4762099]
334000	primes in [2 .. 4777723]
335000	primes in [2 .. 4793237]
336000	primes in [2 .. 4808369]
337000	primes in [2 .. 4823821]
338000	primes in [2 .. 4838963]
339000	primes in [2 .. 4854623]
340000	primes in [2 .. 4869863]
341000	primes in [2 .. 4885339]
342000	primes in [2 .. 4900099]
343000	primes in [2 .. 4915571]
344000	primes in [2 .. 4930963]
345000	primes in [2 .. 4946231]
346000	primes in [2 .. 4961707]
347000	primes in [2 .. 4976899]
348000	primes in [2 .. 4991843]
349000	primes in [2 .. 5007593]
350000	primes in [2 .. 5023307]
351000	primes in [2 .. 5038793]
352000	primes in [2 .. 5054249]
353000	primes in [2 .. 5069663]
354000	primes in [2 .. 5084999]
355000	primes in [2 .. 5100587]
356000	primes in [2 .. 5116141]
357000	primes in [2 .. 5131541]
358000	primes in [2 .. 5146763]
359000	primes in [2 .. 5162387]
360000	primes in [2 .. 5178049]
361000	primes in [2 .. 5193751]
362000	primes in [2 .. 5208887]
363000	primes in [2 .. 5224451]
364000	primes in [2 .. 5239777]
365000	primes in [2 .. 5254859]
366000	primes in [2 .. 5270521]
367000	primes in [2 .. 5286331]
368000	primes in [2 .. 5301481]
369000	primes in [2 .. 5316973]
370000	primes in [2 .. 5332519]
371000	primes in [2 .. 5347789]
372000	primes in [2 .. 5363461]
373000	primes in [2 .. 5378921]
374000	primes in [2 .. 5394401]
375000	primes in [2 .. 5410121]
376000	primes in [2 .. 5425517]
377000	primes in [2 .. 5440889]
378000	primes in [2 .. 5456533]
379000	primes in [2 .. 5472031]
380000	primes in [2 .. 5487701]
381000	primes in [2 .. 5503133]
382000	primes in [2 .. 5518847]
383000	primes in [2 .. 5534677]
384000	primes in [2 .. 5550287]
385000	primes in [2 .. 5566073]
386000	primes in [2 .. 5581397]
387000	primes in [2 .. 5596937]
388000	primes in [2 .. 5612441]
389000	primes in [2 .. 5628257]
390000	primes in [2 .. 5644031]
391000	primes in [2 .. 5659751]
392000	primes in [2 .. 5675381]
393000	primes in [2 .. 5690941]
394000	primes in [2 .. 5706079]
395000	primes in [2 .. 5721481]
396000	primes in [2 .. 5737393]
397000	primes in [2 .. 5752739]
398000	primes in [2 .. 5768663]
399000	primes in [2 .. 5784439]
400000	primes in [2 .. 5800079]
401000	primes in [2 .. 5815519]
402000	primes in [2 .. 5831387]
403000	primes in [2 .. 5846957]
404000	primes in [2 .. 5862617]
405000	primes in [2 .. 5878123]
406000	primes in [2 .. 5893357]
407000	primes in [2 .. 5908739]
408000	primes in [2 .. 5923901]
409000	primes in [2 .. 5939371]
410000	primes in [2 .. 5955031]
411000	primes in [2 .. 5970803]
412000	primes in [2 .. 5986613]
413000	primes in [2 .. 6002371]
414000	primes in [2 .. 6017647]
415000	primes in [2 .. 6033101]
416000	primes in [2 .. 6049411]
417000	primes in [2 .. 6064897]
418000	primes in [2 .. 6080443]
419000	primes in [2 .. 6096271]
420000	primes in [2 .. 6111613]
421000	primes in [2 .. 6127403]
422000	primes in [2 .. 6142987]
423000	primes in [2 .. 6158507]
424000	primes in [2 .. 6174017]
425000	primes in [2 .. 6190007]
426000	primes in [2 .. 6205427]
427000	primes in [2 .. 6221543]
428000	primes in [2 .. 6237019]
429000	primes in [2 .. 6252901]
430000	primes in [2 .. 6268289]
431000	primes in [2 .. 6283583]
432000	primes in [2 .. 6298811]
433000	primes in [2 .. 6314699]
434000	primes in [2 .. 6329927]
435000	primes in [2 .. 6345763]
436000	primes in [2 .. 6361657]
437000	primes in [2 .. 6377773]
438000	primes in [2 .. 6393787]
439000	primes in [2 .. 6409297]
440000	primes in [2 .. 6424937]
441000	primes in [2 .. 6441031]
442000	primes in [2 .. 6456509]
443000	primes in [2 .. 6472309]
444000	primes in [2 .. 6488249]
445000	primes in [2 .. 6503507]
446000	primes in [2 .. 6519329]
447000	primes in [2 .. 6535099]
448000	primes in [2 .. 6550741]
449000	primes in [2 .. 6566363]
450000	primes in [2 .. 6581963]
451000	primes in [2 .. 6597583]
452000	primes in [2 .. 6613001]
453000	primes in [2 .. 6628711]
454000	primes in [2 .. 6644543]
455000	primes in [2 .. 6660721]
456000	primes in [2 .. 6676331]
457000	primes in [2 .. 6692281]
458000	primes in [2 .. 6707717]
459000	primes in [2 .. 6723113]
460000	primes in [2 .. 6738889]
461000	primes in [2 .. 6754537]
462000	primes in [2 .. 6770161]
463000	primes in [2 .. 6786061]
464000	primes in [2 .. 6802121]
465000	primes in [2 .. 6817619]
466000	primes in [2 .. 6833209]
467000	primes in [2 .. 6849061]
468000	primes in [2 .. 6864191]
469000	primes in [2 .. 6879973]
470000	primes in [2 .. 6895393]
471000	primes in [2 .. 6911323]
472000	primes in [2 .. 6927133]
473000	primes in [2 .. 6942721]
474000	primes in [2 .. 6958207]
475000	primes in [2 .. 6973913]
476000	primes in [2 .. 6989813]
477000	primes in [2 .. 7005697]
478000	primes in [2 .. 7021367]
479000	primes in [2 .. 7036819]
480000	primes in [2 .. 7052113]
481000	primes in [2 .. 7068533]
482000	primes in [2 .. 7084097]
483000	primes in [2 .. 7099739]
484000	primes in [2 .. 7115117]
485000	primes in [2 .. 7131629]
486000	primes in [2 .. 7147397]
487000	primes in [2 .. 7163267]
488000	primes in [2 .. 7179101]
489000	primes in [2 .. 7195079]
490000	primes in [2 .. 7210759]
491000	primes in [2 .. 7226431]
492000	primes in [2 .. 7242359]
493000	primes in [2 .. 7258081]
494000	primes in [2 .. 7273759]
495000	primes in [2 .. 7289449]
496000	primes in [2 .. 7305407]
497000	primes in [2 .. 7321091]
498000	primes in [2 .. 7336919]
499000	primes in [2 .. 7353127]
500000	primes in [2 .. 7368787]
501000	primes in [2 .. 7384253]
502000	primes in [2 .. 7400567]
503000	primes in [2 .. 7416313]
504000	primes in [2 .. 7432093]
505000	primes in [2 .. 7447789]
506000	primes in [2 .. 7463921]
507000	primes in [2 .. 7479697]
508000	primes in [2 .. 7495931]
509000	primes in [2 .. 7511969]
510000	primes in [2 .. 7528121]
511000	primes in [2 .. 7543511]
512000	primes in [2 .. 7559173]
513000	primes in [2 .. 7575383]
514000	primes in [2 .. 7591277]
515000	primes in [2 .. 7606561]
516000	primes in [2 .. 7622873]
517000	primes in [2 .. 7638403]
518000	primes in [2 .. 7654139]
519000	primes in [2 .. 7670023]
520000	primes in [2 .. 7685801]
521000	primes in [2 .. 7701523]
522000	primes in [2 .. 7717399]
523000	primes in [2 .. 7733393]
524000	primes in [2 .. 7749493]
525000	primes in [2 .. 7765343]
526000	primes in [2 .. 7781321]
527000	primes in [2 .. 7797563]
528000	primes in [2 .. 7813129]
529000	primes in [2 .. 7829221]
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	I don't know why I generated these primes.  They just seemed so
	beautiful...

    You want to see something beautiful?  Plot the distribution of primes
    with given intervals between them.  In other words, given a set
    of all the primes Pi up to and including Pmax,
    
    	S(k) = count of pairs such that Pj - Pi = k
    
    Note that the count does not require that there be no intervening
    primes.
    
    From the unpublished research of R. G. Steinhardt.

    Another excersize would be to see what th elargest interval between
    primes is.  And how that grows as one calculates more primes.

Along the lines of "how many primes less than <>" (as noted here and in 
topic 812), I'm curious about the average number of prime factors (and 
probably the variability of number of prime factors) for the rest of the 
numbers in the range of interest.  Duplicate prime factors should be counted,
but don't count 1 and the number.  Thus 4 has two prime factors, 12 has
three, 100 has four, etc.  (By this metric primes have no prime factors.
I just figure that the constant of two extra prime factors would skew
the variability stats.)

Next step:  How many distinct factors (more properly, perhaps, divisors?) 
does each number have?  4 has one (2), 12 has four (2, 3, 4, 6), 
100 has seven (2, 4, 5, 10, 20, 25, 50), etc.

Is there anybody still around that has the programs from here and 812
who can use them to answer these questions?

- tom]

    >By this metric primes have no prime factors.
    
    Shouldn't primes have 1 factor rather than none?

        re .-2, .-1
        
        I agree, it seems more natural for the number of prime factors
        (NPF) of a number to be defined as the length of the vector of
        primes you need to multiply together to get the number.  So
        NPF(1) = 0, NPF(2) = NPF(3) = NPF(5) = 1, NPF(4) = NPF(6) = 2,
        etc.
        
        One could also define the number of distinct prime factors of
        a number, where now NDPF(2) = NDPF(4) = NDPF(8) = 1.
        
        I'm not sure what name to give to the function which is zero
        for primes and NPF(n) for composite n.
        
        Dan

>One could also define the number of distinct prime factors of
>a number, where now NDPF(2) = NDPF(4) = NDPF(8) = 1.

Problem 1.  (Probably easy.)  Show that the average of NDPF over the interval
1..n approaches infinity as n does.  Since NPF >= NDPF, this implies the same
for NPF.

Problem 2.  Prove the same for the function f which is 0 at primes and NDPF
elsewhere; hence also for g which is 0 at primes and NPF elsewhere.

    re .4
    
    For a given integer N > 1 with prime factorisation: prod p[i]^k[i]
    where the p[i] are distinct primes, the following hold
    
    (A) NPF = sigma k[i]
    
    (B) NDPF = i
    
    (C) ND = prod (k[i]+1)
    
    where ND is the number of divisors as normally defined. If one wishes
    to exclude 1 and N then just subtract two. (C) is a neat result and is
    reasonably easy to see. Whether any of these results gets one to the
    questions asked I don't know.
    
    e.&o.e.