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Old 2018-07-21, 11:39   #23
mart_r
 
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Quote:
Originally Posted by robert44444uk View Post
The red highlighted totals are the largest proto-gaps for each primorial and are for a given p#, equal to 2*p' where p' is the prime before p in almost all instances.. Hence for 31# the largest proto-gap is 58=29*2, and 29 is the prime before 31. For each case there are 2 such maximum proto-gaps.I think the two are those immediately either side of p#+/-1, and this fact is used in the proof that there are infinitely large gaps between primes given that there are infinite primes.
That would mean the maximum gap would be asymptotic to 2p, which is unfortunately not the case. See "Jacobsthal function".
What you describe is true for p<=7. For p=11, however, there is a gap of 12 (=nextprime(11)-1) before/after 11#-/+1. but a gap of 14 between 113 and 127 (and between 2183 and 2197).

Quote:
Originally Posted by robert44444uk View Post
However, the table shows the curious case of 12 proto-gaps of 40 larger than this for 23#, where the largest proto-gap expected was 38 = 2*19. Can you confirm that this is indeed true and identify one of the 12 proto-gaps n(mod23#) to (n+40)(mod23#)?
The 12 gaps start, respectively, at
20332471
24686821
36068191
65767861
82370089
97689751
125403079
140722741
157324969
187024639
198406009
202760359

For the first instance, 20332471+2*n is divisible by
{1,3,5,11,3,13,23,3,7,19,3,17,5,3,11,7,3,5,13,3,1}
for n=0...20

h(9), the Jacobsthal function for the 9th prime (23), equals 40, so there are no gaps > 40.

Quote:
Originally Posted by robert44444uk View Post
Questions - is 23 the only prime with this property?
Apparently not. For p=37, there are 24 gaps with g=h(12)=66.


P.S.: There's also a way to get to the counts in 41# during the calculation of 37#, which was an overnight-calculation on my PC at my workplace. I'd just have to modify the program a bit. Means, with my current VBA program and resources, I could manage to find the coefficients in my formula for gaps up to 34 or maybe even larger.

Last fiddled with by mart_r on 2018-07-21 at 11:58 Reason: P.S.
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Old 2018-07-21, 18:08   #24
Bobby Jacobs
 
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Quote:
Originally Posted by mart_r View Post
Prime gaps of 2 and 4 have the same probability of occurence because there are at most 2 primes in the interval [p, p+4].
[p, p+6] and larger intervals may contain 3 or more primes, thus cannot have the same number of gaps for any other gap.

There is an interesting way to deduce these numbers, which depend on the admissible prime constellations in the interval [p, p+g]:
Let A(g,p) be the number of gaps of length g between integers in the ring of coprimes to p# (modulo p#), and k the largest number of primes admissible in [p, p+g].
A(g,p)=\sum_{n=1}^{k-1} (a_n * \prod_{primes>n+1} (p-n-1))
where a_n are coefficients depending on g as follows:
Code:
 g   a1    a2     a3     a4   a5   a6
--+----+-----+------+------+----+----
 2    1
 4    1
 6    2    -2
 8    1    -2      1
10  4/3    -3      2
12    2    -7     10     -2
14  6/5    -5   28/3     -3
16    1    -5     12     -6    1
18    2 -23/2  100/3    -22    6
20  4/3 -39/4  116/3    -40   24   -2
22 10/9 -63/8 632/21 -175/6 72/5
24    2  -17 1738/21 -344/3  108  -21
So, for example, with p=23 and g=12, there are 2*7952175 - 7*2867200 + 10*700245 - 2*290304 = 2255792 gaps of length 12 in the ring of coprimes mod 223092870 (=23#).
How do you calculate the coefficients an?
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Old 2018-07-21, 22:55   #25
mart_r
 
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Quote:
Originally Posted by Bobby Jacobs View Post
How do you calculate the coefficients an?

For g=6 and maybe also g=8, I found out by myself, then I looked into the OEIS to get the numbers for g=10, then g=12 found also by some trial-and-error in a spreadsheet, for g=14 and g=16 I let some thousand random numbers do the trial-and-error testing for me, before finally resorting to the more logical approach with linear equation matrices. The fact that a1 is always \prod_{q=3}^p \frac{q-1}{q-2} for q all distinct odd prime factors of g (and a1=1 for all powers of two) is also useful.


Quote:
There's also a way to get to the counts in 41# during the calculation of 37#
This, by the way, only works if the largest gap in pn#, h(n)<2*pn+1. And this is only the case for n<19, i.e. pn<67. Reason: h(19)=152>142, and a gap of 142 in the ring mod 67# has two consecutive numbers that can be cancelled out by the divisor 71.
Could do some more explaining, but it's late at night... maybe tomorrow.

Last fiddled with by mart_r on 2018-07-21 at 22:58
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Old 2018-07-23, 01:25   #26
Bobby Jacobs
 
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Quote:
Originally Posted by mart_r View Post
Quote:
Originally Posted by robert44444uk View Post
The red highlighted totals are the largest proto-gaps for each primorial and are for a given p#, equal to 2*p' where p' is the prime before p in almost all instances.. Hence for 31# the largest proto-gap is 58=29*2, and 29 is the prime before 31. For each case there are 2 such maximum proto-gaps.I think the two are those immediately either side of p#+/-1, and this fact is used in the proof that there are infinitely large gaps between primes given that there are infinite primes.
That would mean the maximum gap would be asymptotic to 2p, which is unfortunately not the case. See "Jacobsthal function".
What you describe is true for p<=7. For p=11, however, there is a gap of 12 (=nextprime(11)-1) before/after 11#-/+1. but a gap of 14 between 113 and 127 (and between 2183 and 2197).
For p=11, the maximum gap is 2*prevprime(11)=2*7=14, so the maximum gap is 2*prevprime(p). However, for p=23, the maximum gap is 40, but 2*prevprime(23)=2*19=38. The gaps with p#+-1 have size nextprime(p)-1.

P.S. I noticed that the quote number for mart_r's post is 492227, which is like the maximal prime gap from 492113 to 492227.

Last fiddled with by Bobby Jacobs on 2018-07-23 at 01:29
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Old 2018-07-23, 07:26   #27
robert44444uk
 
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It seems to me that someone with the right software could create a fractal tree (perhaps similar to a Cantor set) from the iterations of n#, n from 2 to say 17, as each set of n[1]# - however depicted - is one of n[2] subsets within n[2]#. The gaps at the end comb the tree are the only permissible places where primes cannot be, and the width of each gap represents the minimum gap size at that position.

Who is up for it? I don't have software that can do that.
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Old 2018-07-26, 11:32   #28
robert44444uk
 
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Quote:
Originally Posted by robert44444uk View Post
It seems to me that someone with the right software could create a fractal tree (perhaps similar to a Cantor set) from the iterations of n#, n from 2 to say 17, as each set of n[1]# - however depicted - is one of n[2] subsets within n[2]#. The gaps at the end comb the tree are the only permissible places where primes cannot be, and the width of each gap represents the minimum gap size at that position.

Who is up for it? I don't have software that can do that.
The most obvious shape to draw would be a series of concentric circles, each representing the modular clock for each primorial iteration with some clock hands showing the allowed and forbidden positions for primes.. You could imagine overlaying clock hands for higher primes at various mods that would show how those hands interfere with the gaps between the existing hands.

Last fiddled with by robert44444uk on 2018-07-26 at 11:36
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Old 2018-07-27, 00:29   #29
rudy235
 
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I was looking at how the prime gaps vary when one goes from A) m*210 ~(m+1)*210
to B) m*2310 ~m+1(2310)

At the levels of A) the 4 and 2 dominate and 6 is a close 3rd
Of 48 gaps #2 and #4 tie at 15 each, #6 reaches 14 and #8 and #10 are way below with 2 each. (15+15+14+2+2) =48

When you transition to B) there are supposedly 528 gaps (48 *11) but exactly 48 multiples of 11 are eliminated. As a Result 96 GAPS dissappear -2 for each multiple of 11- and more importantly 48 NEW gaps appear.

So who loses and who gains?
Main losers are #2 and #4 with 29 and 31 loses.
#6 gains 15 but loses 29 with a net loss of 14
#10 gains 12 and loses 3 with a net gain of 9
#8 gain 8; #12 gain 7 and #14 gains 2

Total loses are (as expected) 48 gaps

Now going a little further on;

If we transition from B) to m*30030 ~ (m+1)*30030 there will be 528 multiples of 13 which mean a net loss of 528 gaps.

#2 and #4 gaps will lose more. (there are no possible gains for them, only loses)
#6 will lose probably around half of what each the previous numbers lose..
And 10, 12, 14 and 18 will appear more frequently.They will have a net gain.

What is clear is that 2 and 4 will always lose proportionally more. The reason is very simple. They cannot gain a new gap. The can only lose gaps.

6, 8, 10 and 12 can always get new gaps.
Code:
A 2 and a 4 makes a 6
A 2 and a 6 makes an 8 (also a 4 and 4 makes an 8)
A 4 and a 6 makes a 10 (also a 2 and an 8 makes a 10)
An 8 and a 4 makes a 12 (also a 6 and a 6, or, a 10 and 2)

Last fiddled with by rudy235 on 2018-07-27 at 01:18 Reason: clarifying a bit.
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Old 2018-07-27, 12:00   #30
robert44444uk
 
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It is always good to have a race, I'm not sure if this sequence is well known, but clearly primes can only be certain values mod2310 and hence occasionally there is going to be one value x(mod2310) which has more primes up to that point than any other value mod2310. Lets call that x the champion at that level of prime. The champion remains the champion until deposed, i.e. there may be other values mod2310 with the same number of primes as the champion.


Looking at all primes up to 1e9, we have the following interesting table:

A = the prime at which the value of the champion x changes
B = the value of x
C = the number of primes up to A which are x(mod2310)

Code:
A       B       C
2	2	1
2333	23	2
6959	29	4
9277	37	5
13931	71	7
17189	1019	8
21727	937	9
26111	701	10
28493	773	11
44797	907	15
49117	607	16
58657	907	18
127423	373	32
136949	659	35
143593	373	36
183971	1481	43
190657	1237	44
194809	769	45
205963	373	47
210143	2243	48
224977	907	51
230933	2243	52
234527	1217	53
238837	907	54
242483	2243	55
247249	79	56
265007	1667	59
275719	829	61
409777	907	85
423559	829	87
473659	109	96
478999	829	97
520549	799	103
529819	829	104
569927	1667	111
580639	829	113
599611	1321	116
606049	829	117
618833	2063	120
661459	799	125
690233	1853	129
696109	799	131
729977	17	138
1026313	673	186
1047821	1391	188
1054033	673	189
1094957	17	196
1104853	673	198
1130273	683	202
1166567	17	208
1215719	659	216
1231247	17	217
1307281	2131	229
1360171	1891	238
1388243	2243	242
1393589	659	243
1415611	1891	247
1440403	1273	250
1450697	17	252
1471403	2243	256
1505083	1273	260
1511423	683	261
1515377	17	262
1525873	1273	264
1552337	17	268
1559183	2243	269
1561577	17	270
1587653	683	274
1603799	659	277
1610753	683	278
1613653	1273	279
2477641	1321	407
2484523	1273	408
2546941	1321	418
2553823	1273	419
2676301	1321	436
2701663	1273	438
2821831	1321	457
2842573	1273	460
2854171	1321	462
2872027	697	463
3029731	1321	489
3195427	697	511
3214531	1321	513
3271657	697	519
3329609	899	527
3380213	683	534
3387157	697	535
3395101	1711	536
3407947	697	539
3426413	683	543
3478261	1711	548
3484177	697	549
3493403	683	550
3502657	697	552
3671273	683	575
3675907	697	576
3694373	683	581
4155517	2137	641
4174853	683	643
4192477	2137	647
4232603	683	652
4312103	1643	662
4319351	1961	663
4325593	1273	664
4328273	1643	665
4517323	1273	692
4520003	1643	693
4561213	1273	697
4623953	1643	705
4629913	673	706
4635503	1643	707
4649681	1961	708
4658281	1321	709
4663541	1961	710
4700183	1643	716
4716671	1961	718
4804627	2137	733
4813691	1961	734
4852643	1643	737
4864511	1961	738
4884983	1643	742
4887611	1961	743
4979693	1643	757
4993871	1961	759
5009723	1643	761
5023901	1961	763
5092883	1643	772
5109371	1961	774
5137373	2243	778
5213321	1961	788
5215913	2243	789
5252591	1961	794
5303693	2243	800
5310763	73	802
5315243	2243	803
5321891	1961	804
5326793	2243	805
5398543	73	816
5409953	2243	817
5442509	149	822
5670701	1961	853
5735879	149	862
5770031	1961	866
5777459	149	868
5797751	1961	870
5815133	863	871
5821349	149	872
5879813	863	880
5906801	131	884
5909843	863	885
5927591	131	886
5932943	863	887
5941451	131	888
5957639	149	891
5971481	131	894
6007961	1961	899
6066659	599	906
6073079	89	907
6084671	131	909
6093869	89	910
6098531	131	911
6297209	149	937
6324911	131	941
6331859	149	942
6391901	131	949
6420173	683	953
6424241	131	954
6434033	683	956
6442721	131	957
6459443	683	958
6475061	131	960
6510263	683	965
6581387	197	976
6593423	683	977
6862039	1339	1012
6884483	683	1014
6889759	1339	1015
6900281	311	1017
7216589	149	1065
7316453	683	1078
7392149	149	1088
7692611	311	1129
7703999	149	1131
7713401	311	1132
7717859	149	1133
7748051	311	1137
7794089	149	1142
7796561	311	1143
7803329	149	1144
7821971	311	1148
7840289	149	1150
7858931	311	1152
7909589	149	1158
8387921	311	1216
8394689	149	1217
8406193	103	1218
8420261	311	1220
8805869	149	1275
8833751	311	1278
8863619	149	1281
8870711	311	1283
8895959	149	1286
8983901	311	1299
8995289	149	1301
9004691	311	1302
9020699	149	1304
9177941	311	1322
9200879	149	1325
9972581	311	1427
9997829	149	1430
10192031	311	1455
10221899	149	1459
10231301	311	1461
10272719	149	1466
10295981	311	1469
10302749	149	1470
10328711	701	1474
10351259	149	1476
10363361	701	1478
10376669	149	1480
10386071	311	1482
10445969	149	1487
10484249	1469	1491
10508339	149	1496
10562891	1571	1502
10599749	1469	1507
10602161	1571	1508
10627469	1469	1509
10723169	149	1522
10730297	347	1524
10865167	1237	1541
10900963	73	1546
10915987	1237	1548
10922027	347	1549
10955489	1469	1555
10961297	347	1556
11311481	1721	1602
11333009	149	1603
11336891	1721	1604
11395577	347	1612
11399261	1721	1613
11753353	73	1661
11804249	149	1669
13106351	1721	1840
13139429	149	1846
13214921	1721	1858
13220279	149	1859
13258811	1721	1863
13284959	149	1866
13700021	1721	1918
13746959	149	1923
13771631	1721	1926
14720401	1081	2048
14723351	1721	2049
14750431	1081	2051
14848753	73	2063
14859001	1081	2064
14864261	1721	2065
14943463	73	2079
14947421	1721	2080
15006841	1081	2084
15038173	73	2088
15041491	1081	2089
15068203	73	2092
15071521	1081	2093
15390941	1721	2134
15434191	1081	2138
15460241	1721	2140
15478081	1081	2141
15497201	1721	2143
15503491	1081	2144
15531851	1721	2148
15540451	1081	2149
15550331	1721	2150
15554311	1081	2151
15571121	1721	2152
15612061	1081	2159
16314941	1721	2245
16519891	1081	2270
16562111	1721	2276
16599959	299	2280
16737671	1721	2298
16784759	299	2305
16790801	1721	2306
17206201	1321	2358
17271281	1721	2368
17298601	1321	2373
17347511	1721	2377
17721331	1321	2426
17793341	1721	2435
19316519	299	2625
19320251	1721	2626
19328069	299	2627
19345661	1721	2628
19501319	299	2645
19523947	2137	2648
19563689	299	2652
19627897	2137	2660
19713839	299	2671
20244173	1643	2734
20272859	299	2739
20276513	1643	2740
20344469	299	2750
20368913	1643	2752
20410987	2137	2759
20424353	1643	2760
20468737	2137	2763
20516753	1643	2769
20653537	2137	2785
20713103	1643	2792
20778787	337	2802
20789333	1643	2804
20852707	337	2811
21024221	911	2831
21081397	337	2837
21094747	2137	2839
21136837	337	2845
21157117	2137	2846
21199207	337	2853
21330367	2137	2869
21353977	337	2873
21395047	2137	2878
21409417	337	2879
23251061	911	3112
23266657	337	3114
23355011	911	3126
23368297	337	3128
23371181	911	3129
23419117	337	3138
23424311	911	3139
23456077	337	3144
23694581	911	3170
24609887	1457	3285
24613961	911	3286
24656087	1457	3291
24803899	1429	3306
24820423	1783	3308
24827027	1457	3309
24850453	1783	3311
24981797	1457	3330
24993673	1783	3331
24997421	911	3332
25007533	1783	3334
25025141	911	3336
25030633	1783	3337
25075961	911	3342
25090693	1783	3343
25101371	911	3344
26249987	1457	3482
26254061	911	3483
26298497	1457	3487
26304881	911	3488
26744327	1457	3542
26787671	911	3546
26843657	1457	3551
26875451	911	3557
26880617	1457	3558
26896241	911	3560
27036721	481	3580
27055631	911	3582
27156841	481	3595
27161891	911	3596
29598511	481	3890
29610883	1303	3891
29621611	481	3892
29629363	1303	3893
29652071	911	3896
29756383	1273	3908
29776811	911	3910
29781823	1303	3911
29795291	911	3913
29824853	443	3915
29864591	911	3922
29898773	443	3924
29901913	1273	3925
29915411	911	3927
29975833	1273	3933
29998571	911	3935
30005863	1273	3936
30015133	1303	3937
30023551	481	3938
30077111	911	3945
30176803	1273	3958
30201851	911	3962
30222211	481	3964
30232243	1273	3965
30236071	481	3966
30246103	1273	3968
30293821	481	3973
30434291	41	3995
30462451	481	3998
30492041	41	4003
30610291	481	4017
30632951	41	4020
30864209	299	4050
30868571	41	4051
31980083	443	4191
31995851	41	4193
32000873	443	4194
32030501	41	4199
32051693	443	4201
32136761	41	4211
32213393	443	4221
32233781	41	4223
32386643	443	4244
32423201	41	4248
32432843	443	4249
32462471	41	4253
33222863	443	4347
33245561	41	4349
33255203	443	4350
33282521	41	4353
33359153	443	4364
33363371	41	4365
33368393	443	4366
33418811	41	4371
33673313	443	4398
33686771	41	4399
33719513	443	4402
33855401	41	4417
33860423	443	4418
33864641	41	4419
33934343	443	4427
33975521	41	4430
33978233	443	4431
34019411	41	4439
34035983	443	4442
34109501	41	4451
34112213	443	4452
34125671	41	4453
34637671	1531	4514
34643111	41	4515
35678653	703	4641
35698781	41	4643
35777723	443	4653
35825831	41	4660
36995353	703	4800
37054751	41	4810
37060033	703	4811
37087091	41	4814
37242523	703	4836
37248791	41	4837
37272553	703	4839
37338881	41	4846
37374193	703	4850
37398941	41	4853
37494313	703	4863
37498271	41	4864
37554373	703	4872
37558331	41	4873
37579783	703	4875
37636871	41	4881
37699903	703	4886
37720031	41	4889
37734553	703	4892
37747751	41	4894
37783063	703	4898
38356793	1553	4967
38381353	703	4968
38719463	1553	5011
38873383	703	5027
38892713	1553	5030
38908033	703	5032
39213803	1553	5067
39224503	703	5068
39322373	1553	5078
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204834937	307	23795
204841141	1891	23796
204862657	307	23798
204874819	919	23800
204904237	307	23803
206019241	1891	23924
206033827	307	23926
206601361	1891	23985
206606707	307	23986
207061051	1891	24035
207066397	307	24036
207349801	1891	24069
207389797	307	24074
207756361	1891	24115
207766327	307	24116
207777151	1891	24117
207780187	307	24118
209134217	677	24265
209420287	307	24300
209538467	677	24313
209561197	307	24315
209580047	677	24317
209593537	307	24320
210005087	677	24361
210434377	307	24405
210441677	677	24406
210452857	307	24407
210476327	677	24410
210489817	307	24411
210517907	677	24414
210549877	307	24418
210598757	677	24425
210616867	307	24427
210633407	677	24428
211143547	307	24484
211167017	677	24487
211182817	307	24488
211185497	677	24489
211198987	307	24490
211259417	677	24497
211279837	307	24498
211384157	677	24509
211390717	307	24510
211451147	677	24517
212245417	307	24617
212273507	677	24620
212480311	1891	24642
212539157	677	24649
212548027	307	24650
212559947	677	24651
213051607	307	24704
213869717	677	24794
213980227	307	24807
214003697	677	24809
214019497	307	24810
263674549	1909	30212
263684497	307	30213
264034909	1909	30259
264044857	307	30260
264258979	1909	30284
264271237	307	30285
264284389	1909	30287
264335917	307	30292
336721213	1753	38017
336747487	307	38020
337035373	1753	38052
337137877	307	38059
337153183	1753	38061
343913107	307	38767
343928413	1753	38769
344157967	307	38794
344265673	1753	38809
344282707	307	38812
345464563	1753	38937
345467737	307	38938
345471493	1753	38939
352159807	307	39657
352200523	1753	39660
352203697	307	39661
352235173	1753	39664
353742157	307	39820
353752843	1753	39821
354308107	307	39879
354944803	1753	39942
380066209	1909	42593
380072983	1753	42594
380086999	1909	42595
380211583	1753	42611
380234839	1909	42613
380262403	1753	42615
380274109	1909	42617
380297053	1753	42619
380299519	1909	42620
380303983	1753	42621
380673739	1909	42659
380708233	1753	42662
380715319	1909	42663
380867623	1753	42678
381082609	1909	42701
381100933	1753	42702
381105709	1909	42703
383071363	1753	42908
383087689	1909	42910
383152213	1753	42919
383173159	1909	42920
383205343	1753	42922
383524279	1909	42956
383540293	1753	42957
383776069	1909	42976
383780533	1753	42977
383799877	307	42979
383879863	1753	42986
384258859	1909	43023
384432817	307	43042
384469069	1909	43046
384633787	307	43060
384686209	1909	43063
384730807	307	43067
384739339	1909	43068
384753907	307	43071
384986509	1909	43097
385157293	1753	43123
385178239	1909	43126
385228903	1753	43132
385335319	1909	43141
385339783	1753	43142
385351489	1909	43143
386812117	307	43299
386864539	1909	43304
386883727	307	43306
386887639	1909	43307
386890657	307	43308
386950009	1909	43316
386957647	307	43317
387781609	1909	43398
420467023	823	46847
420710659	1909	46872
420725743	823	46873
420925489	1909	46892
420986773	823	46900
420997099	1909	46901
421000633	823	46902
421054849	1909	46907
421067623	823	46908
421121839	1909	46913
421915393	823	47002
421923409	1909	47003
421966213	823	47008
422055079	1909	47018
422060923	823	47019
422092039	1909	47021
422116363	823	47023
422142859	1909	47026
422153323	823	47027
422214469	1909	47034
422280373	823	47041
422422369	1909	47057
422513683	823	47065
422521699	1909	47066
422543713	823	47069
423940039	1909	47233
424054453	823	47245
425737219	1909	47424
427025113	823	47566
427120909	1909	47578
427685773	823	47638
427691479	1909	47639
427986073	823	47667
428012569	1909	47670
428016103	823	47671
428072629	1909	47676
428103883	823	47678
428160409	1909	47684
428177803	823	47686
428195059	1909	47688
428198593	823	47689
428232019	1909	47691
428244793	823	47692
439206829	1909	48845
439210363	823	48846
439320019	1909	48857
439463083	1753	48873
439467859	1909	48874
439476943	1753	48875
439484029	1909	48876
439502353	1753	48878
439509439	1909	48879
445970353	1753	49565
445988989	1909	49566
446067373	1753	49573
446180719	1909	49583
454364893	1753	50436
454374289	1909	50438
454378753	1753	50439
454408939	1909	50442
454794553	1753	50486
454840909	1909	50491
454859233	1753	50492
455289049	1909	50539
455305063	1753	50540
455316769	1909	50542
455406703	1753	50552
455473849	1909	50556
455861773	1753	50595
455891959	1909	50599
455924143	1753	50601
455928919	1909	50602
463032013	1753	51334
463043719	1909	51336
463119793	1753	51346
463122259	1909	51347
463133653	1753	51348
463161529	1909	51349
463170613	1753	51350
464064739	1909	51445
464080753	1753	51446
464462059	1909	51487
464468833	1753	51488
464487469	1909	51491
464521963	1753	51493
464702299	1909	51515
464870773	1753	51533
464917129	1909	51536
464930833	1753	51538
464937919	1909	51539
464981653	1753	51542
465605509	1909	51607
465616903	1753	51609
465621679	1909	51610
465626143	1753	51611
465757969	1909	51624
465771673	1753	51625
465785689	1909	51627
465926443	1753	51641
466088299	1909	51654
466254463	1753	51672
467100079	1909	51762
468883243	1753	51951
468936529	1909	51957
468940993	1753	51958
470791459	1909	52160
470798233	1753	52161
470819179	1909	52164
470828263	1753	52165
481301959	1909	53245
481417303	1753	53258
481583779	1909	53275
481592863	1753	53276
481604569	1909	53277
482634673	1753	53380
483937669	1909	53519
483955993	1753	53521
484109317	307	53539
484131553	1753	53542
575191867	1867	63005
575198683	1753	63006
575210347	1867	63008
575233333	1753	63012
575240377	1867	63014
575385793	1753	63029
575392837	1867	63030
575427529	1909	63034
575436727	1867	63035
575446009	1909	63038
575462137	1867	63039
576344599	1909	63127
576427717	1867	63134
576430069	1909	63135
576441577	1867	63136
576508453	1753	63141
576513187	1867	63142
576524623	1753	63144
576531667	1867	63145
576894379	1909	63183
576919747	1867	63186
576933649	1909	63187
576940537	1867	63188
577201609	1909	63217
577474147	1867	63246
577781419	1909	63277
577792927	1867	63278
577901539	1909	63292
578130073	1753	63314
578134849	1909	63315
578141623	1753	63316
578221027	307	63325
578310253	1753	63335
578618347	307	63367
578629033	1753	63369
579149647	307	63419
579185743	1753	63423
579235117	307	63428
579460633	1753	63449
579481579	1909	63451
579600097	307	63461
579670999	1909	63470
579710977	307	63473
579726439	1909	63476
579757177	307	63480
579791119	1909	63485
580015897	307	63511
580527853	1753	63566
580551817	307	63568
580733599	1909	63586
580775887	307	63591
580823689	1909	63597
580896007	307	63604
580902229	1909	63605
581009197	307	63614
581449699	1909	63665
581471197	307	63667
581477419	1909	63668
581662927	307	63688
583260739	1909	63853
583286857	307	63855
583295389	1909	63856
583319197	307	63858
583364689	1909	63863
583605637	307	63888
583623409	1909	63889
583721137	307	63900
583773559	1909	63904
583790437	307	63907
585792499	1909	64119
586679497	1867	64204
586767319	1909	64212
586806547	1867	64218
586811209	1909	64220
589793377	1867	64521
589800349	1909	64522
590072887	1867	64550
590112199	1909	64555
590116777	1867	64556
590130679	1909	64557
590419387	1867	64589
590472559	1909	64596
591017677	1867	64651
591121669	1909	64660
591232507	1867	64673
591262579	1909	64677
592059487	1867	64764
592084939	1909	64767
592089517	1867	64768
592096489	1909	64769
592117237	1867	64772
592452229	1909	64805
592486837	1867	64810
592558489	1909	64818
592572307	1867	64819
592576969	1909	64820
592593097	1867	64823
592600069	1909	64824
592620817	1867	64825
605332789	1909	66152
605341987	1867	66153
605353579	1909	66155
605378947	1867	66156
605404399	1909	66159
605420527	1867	66161
605445979	1909	66164
605462107	1867	66166
605536069	1909	66172
605545267	1867	66173
605621539	1909	66181
605644597	1867	66183
605681599	1909	66187
605686177	1867	66188
605716249	1909	66191
605730067	1867	66192
605755519	1909	66194
606062707	1867	66229
606067369	1909	66230
606238267	1867	66249
606275269	1909	66252
606302947	1867	66255
606314539	1909	66256
606550117	1867	66287
606557089	1909	66288
606714127	1867	66306
607007539	1909	66337
607039837	1867	66341
607079149	1909	66344
607097587	1867	66346
607178479	1909	66354
607194607	1867	66357
607421029	1909	66378
607455637	1867	66381
607485709	1909	66386
607506457	1867	66387
893069189	89	95561
893357407	1867	95590
893367179	89	95592
893412847	1867	95597
893422619	89	95599
893435947	1867	95600
895494689	89	95806
895505707	1867	95807
895570919	89	95813
895676647	1867	95821
895734929	89	95828
895745947	1867	95829
896645069	89	95922
896663017	1867	95924
896684339	89	95925
896785447	1867	95936
916039829	89	97879
916087807	1867	97884
916122989	89	97888
916279537	1867	97899
916307789	89	97902
916325737	1867	97903
916848329	89	97953
916861657	1867	97954
916864499	89	97955
916891687	1867	97959
916896839	89	97960
916974847	1867	97968
916989239	89	97970
919562047	1867	98217
919581059	89	98219
919633657	1867	98227
919738139	89	98235
919756087	1867	98237
Not all permissible values of x appear in the table - the following 85 out of 480 permissible values appear up to prime closest to 1e9.
D = the value x
E = the number of times x appears as the leading mod2310 value for all primes up to 1e9

Code:
D       E
1867	41510
307	18117
1753	15978
1909	8055
601	5487
919	2111
2273	1885
823	1633
787	1500
643	1234
1553	845
41	771
911	716
149	616
1721	579
677	551
2279	481
1891	468
2201	369
89	348
337	319
703	250
821	240
1273	210
1357	194
1217	177
1643	166
1081	164
683	155
311	152
299	99
443	91
1961	86
17	81
551	80
347	70
697	70
1321	61
481	59
2137	55
829	52
131	50
2243	45
1457	33
1783	29
1469	20
907	19
73	16
673	14
799	12
373	11
1189	10
2131	9
659	7
863	7
899	7
1237	7
1303	7
1787	7
1571	6
701	5
2063	5
773	4
1339	4
1667	4
1711	4
79	3
23	2
37	2
103	2
607	2
743	2
769	2
1429	2
1853	2
29	1
71	1
109	1
197	1
599	1
937	1
1019	1
1391	1
1481	1
1531	1

Last fiddled with by robert44444uk on 2018-07-27 at 12:13
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Old 2018-07-27, 17:23   #31
rudy235
 
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Jun 2015
Vallejo, CA/.

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Quote:
Originally Posted by robert44444uk View Post
Not all permissible values of x appear in the table - the following 85 out of 480 permissible values appear up to prime closest to 1e9.
D = the value x
E = the number of times x appears as the leading mod2310 value for all primes up to 1e9

Code:
D       E
1867	41510
307	18117
[...]
Perhaps you could submit this number 1867 to this page.https://primes.utm.edu/curios/page.php/1867.html


Any idea if this number remains as the most popular at levels of 10^10 or 10^11?
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Old 2018-07-29, 08:04   #32
robert44444uk
 
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Jun 2003
Suva, Fiji

23×3×5×17 Posts
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Quote:
Originally Posted by rudy235 View Post
Perhaps you could submit this number 1867 to this page.https://primes.utm.edu/curios/page.php/1867.html


Any idea if this number remains as the most popular at levels of 10^10 or 10^11?
I doubt this demonstrates important enough curio quality for 1867 due to the law of small numbers. I plan to run this series up to 1e11 when I return home. Travelling at present.

I wonder if each permissible x appears once in the second table if taken out to infinity. Will have to think more about that.

Last fiddled with by robert44444uk on 2018-07-29 at 08:07
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Old 2018-07-29, 13:21   #33
Bobby Jacobs
 
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May 2018

271 Posts
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We should try primes mod n for different values of n.
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