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 2022-03-06, 12:23 #1 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×7×263 Posts (conjectured or proved) finite-record sequences only list the base-independent sequences, thus for example, least start of exactly n consecutive numbers that are harshad numbers (A060159) is not listed here. A343816: least start of exactly n consecutive odd numbers that are cyclic numbers: {23, 41, 177, 1, 11, 877, 2387, 695} (no further terms, since numbers divisible by 9 cannot be cyclic number) A138496: record number of occurrences of n as an entry in rows <= n of Pascal's triangle: {0, 1, 10, 120, 3003}, or {2, 3, 6, 10, 120, 3003} (if the number 1 (which appear infinitely many times in Pascal's triangle) is not counted, see A062527), is there a number (>1) appearing >8 times in Pascal's triangle? {6, 13, 454} with difference {25-6=19, 35-27=8, 602-596=6} (see A152454, record with smaller difference in a single row of A152454), is there a next row with a smaller difference? {0, 2, 383, 10103}, see A000790, with record values {4, 341, 382, 561}, also for the negative bases, {0, 1, 13, 22, 58, 277, 877, 1822, 5098, 23602}, with record values {4, 9, 14, 21, 57, 69, 219, 451, 481, 561} (no further terms, since 561 is a Carmichael number, thus 561 is weak pseudoprime to every (positive or negative or 0) base) Record of the smallest n>=1 such that k*2^n+1 is prime for odd k: {1, 7, 17, 19, 31, 47, 383, 2897, 3061, 4847, 5359, 10223, 21181, ..., 78557}, the record values are {1, 2, 3, 6, 8, 583, 6393, 9715, 33288, 3321063, 5054502, 31172165, ..., infinity}, since 78557 is Sierpinski number, there is no n>=1 such that 78557*2^n+1 is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Record of the smallest n>=1 such that 2^n+k is prime (or PRP) for odd k: {1, 7, 23, 31, 47, 61, 139, 271, 287, 773, 2131, 40291, 78557}, the record values are {1, 2, 3, 4, 5, 8, 10, 20, 29, 955, 4583176, 9092392, infinity}, since 78557 is Sierpinski number, there is no n>=1 such that 2^n+78557 is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Record of the smallest n>=1 such that k*2^n-1 is prime for odd k: {1, 13, 23, 43, 59, 127, 191, 659, 2293, 23669, ..., 509203}, the record values are {2, 3, 4, 7, 12, 25, 226, 800516, 12918431, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that 509203*2^n-1 is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Record of the smallest n>=1 such that 2^n-k is prime (or PRP) for odd k: {1, 3, 7, 127, 221, 337, 1061, 1871, ..., 509203}, the record values are {2, 3, 39, 47, 714, 791, 970, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that 2^n-509203 is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Record of the smallest n>=1 such that |2^n-k| is prime (or PRP) for odd k: {1, 3, 29, 127, 337, 2293, ..., 509203}, the record values are {2, 3, 4, 47, 791, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that |2^n-509203| is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Primes p such that the absolute value of the fraction A260209(A000720(p)) / (p^3) is a record low: {2, 3, 11, 13, 31, 103, 211, 2851, 12101, 16843}, or start with the prime 5, {5, 11, 13, 31, 103, 211, 2851, 12101, 16843} (no further terms, since for the prime 16843, the value is 0) Last fiddled with by sweety439 on 2022-05-04 at 02:35
 2022-03-06, 13:44 #2 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×7×263 Posts A291466: least start of exactly n consecutive numbers that are sphenic numbers: {30, 230, 1309} (no further terms, since numbers divisible by 4 cannot be sphenic numbers) least start of exactly n consecutive numbers that are Achilles numbers: {72, 5425069447, {105, 66, 406} least start of exactly n consecutive triangular numbers that are sphenic numbers (no further terms, since if 4 consecutive triangular numbers are all sphenic numbers, let them be n*(n+1)/2, (n+1)*(n+2)/2, (n+2)*(n+3)/2, (n+3)*(n+4)/2, then we have omega(n*(n+1)) = omega((n+1)*(n+2)) = omega((n+2)*(n+3)) * omega((n+3)*(n+4) = 4, and none of n, n+1, n+2, n+3, n+4 is divisible by 8 (or at least one of n*(n+1)/2, (n+1)*(n+2)/2, (n+2)*(n+3)/2, (n+3)*(n+4)/2 will be divisible by 4 and cannot be sphenic number), thus omega(n) = omega(n+2) = omega(n+4), but n, n+2, n+4 cannot be all primes unless n=3, thus omega(n) must be >= 2, and if omega(n) = 2, then we have omega(n+1) = 2 (since omega(n*(n+1)) = 4), similarly, omega(n+2) = omega(n+3) = omega(n+4) = 2, which is impossible since at least one of n, n+1, n+2, n+3, n+4 is divisible by 4, thus this number can only be 4, thus omega(n) must be 3, and omega(n+1) = omega(n+3) = 1, i.e. n+1 and n+3 are twin primes, thus neither n+1 nor n+3 is divisible by 3, thus n+2 is divisible by 3, and n+2 cannot be divisible by 12 since omega(n+2) = 3 and if n+2 is divisible by 12 then n+2 can only be 12, thus n and n+4 must be divisible by 4 (note that neither n+1 nor n+3 cannot be divisible by 4 since n+1 and n+3 are twin primes), and thus one of n and n+4 will be divisible by 8, which is a contradiction! The smallest prime factor of A005165(n), this record sequence is finite since for all n>=3612702, the smallest prime factor of A005165(n) are all 3612703, but we do not know the last record n and its record value (since this requires to factor near 22122513-digit numbers, the last record n and its record value will not be known in our lifetime! but we know that they really exist! this is equivalent to this problem: Find the smallest m such that gcd(A005165(n),m) > 1 for all n >= 2, such m must exist, but we cannot find the m in our lifetime) Last fiddled with by sweety439 on 2022-03-06 at 13:59
 2022-04-24, 20:29 #3 sweety439   "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 368210 Posts Related page: What is special about this number? (Some numbers, like 43 and 94, breaks a law of small numbers, also some numbers, like 24 and 41, are the (proved) largest number satisfying a property.
2022-05-14, 13:36   #4
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

2·7·263 Posts

Quote:
 Originally Posted by sweety439 Record of the smallest n>=1 such that k*2^n+1 is prime for odd k: {1, 7, 17, 19, 31, 47, 383, 2897, 3061, 4847, 5359, 10223, 21181, ..., 78557}, the record values are {1, 2, 3, 6, 8, 583, 6393, 9715, 33288, 3321063, 5054502, 31172165, ..., infinity}, since 78557 is Sierpinski number, there is no n>=1 such that 78557*2^n+1 is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Record of the smallest n>=1 such that 2^n+k is prime (or PRP) for odd k: {1, 7, 23, 31, 47, 61, 139, 271, 287, 773, 2131, 40291, 78557}, the record values are {1, 2, 3, 4, 5, 8, 10, 20, 29, 955, 4583176, 9092392, infinity}, since 78557 is Sierpinski number, there is no n>=1 such that 2^n+78557 is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Record of the smallest n>=1 such that k*2^n-1 is prime for odd k: {1, 13, 23, 43, 59, 127, 191, 659, 2293, 23669, ..., 509203}, the record values are {2, 3, 4, 7, 12, 25, 226, 800516, 12918431, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that 509203*2^n-1 is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Record of the smallest n>=1 such that 2^n-k is prime (or PRP) for odd k: {1, 3, 7, 127, 221, 337, 1061, 1871, ..., 509203}, the record values are {2, 3, 39, 47, 714, 791, 970, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that 2^n-509203 is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Record of the smallest n>=1 such that |2^n-k| is prime (or PRP) for odd k: {1, 3, 29, 127, 337, 2293, ..., 509203}, the record values are {2, 3, 4, 47, 791, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that |2^n-509203| is prime (or we can say this n is infinity), thus of course there cannot be any further terms.
We use "infinity" instead of "0" or "-1" because ...

* infinity is > any finite number, this is more convenient (e.g. the smallest n such that (k^n-1)/(k-1) is prime, for k = 185, (k^n-1)/(k-1) has been searched to n=66337 with no prime or PRP found, we can use ">66337" for k = 185, ">66337" includes infinity (since infinity is > 66337) but does not includes 0 or -1, it is still possible that there is no n such that (k^n-1)/(k-1) is prime for k = 185)
* "the smallest n such that (k^n-1)/(k-1) is prime" is "the largest n such that (k^r-1)/(k-1) is not prime for all r<n", for the case that (k^n-1)/(k-1) is never prime (k = 9, 25, 32, 49, 64, 81, 121, 125, 144, ...), such n is infinity instead of 0 or -1
* let S(k) = {n: (k^n-1)/(k-1) is prime}, a(n) = min(S(k)), then a(n) = the smallest n such that (k^n-1)/(k-1) is prime, and S(2) = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, ...}, a(2) = 2, S(3) = {3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867, ...}, a(3) = 3, S(8) = {3}, a(8) = 3, S(9) = {} (the empty set), a(9) = infinity (due to the definition of "sup" and "inf", sup(empty set) = -infinity and inf(empty set) = +infinity) instead of 0 or -1
* A058887(n) is the smallest k such that A057192(k) > n, A058887(n) <= 271129 for all n, and A057192(271129) should be infinity instead of 0 or -1

(but the only technical problem is that OEIS sequences cannot use "infinity" thus in OEIS sequence we can use "0" or "-1" to replace "infinity", such as A084740, a(9), a(25), a(32), a(49), a(64), etc. should be "infinity" instead of "0", also A040076(78557) and A040081(509203), should be "infinity" instead of "-1", but an example is A001438, which a(0) = a(1) = infinity

Also see http://gladhoboexpress.blogspot.com/...-derbread.html and http://chesswanks.com/seq/a306861.txt and http://chesswanks.com/seq/a269254.txt

Last fiddled with by sweety439 on 2022-12-01 at 15:41

2022-12-08, 23:27   #5
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

71428 Posts

Quote:
 Originally Posted by sweety439 We use "infinity" instead of "0" or "-1" because ...
Another example of "infinity" in number theory:

Since for all positive integer n we have 1^(p-1) == 1 mod p^n

Code:
 Basis  Primzahlen^2 (P^2)
1   All primes^infinity
2   1093, 3511
3   11, 1006003
4   1093, 3511
5   2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
6   66161, 534851, 3152573
7   5, 491531
8   3, 1093, 3511
9   2^3, 11, 1006003
10   3, 487, 56598313
11   71
12   2693, 123653
13   2, 863, 1747591
14   29, 353, 7596952219
15   29131, 119327070011
16   1093, 3511
17   2^4, 3, 46021, 48947, 478225523351
18   5, 7^3, 37, 331, 33923, 1284043
19   3, 7^3, 13, 43, 137, 63061489
20   281, 46457, 9377747, 122959073
21   2
22   13, 673, 1595813, 492366587
23   13, 2481757, 13703077, 15546404183
24   5, 25633
25   2^3, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
26   3^3, 5, 71, 486999673, 6695256707
27   11, 1006003
28   3^3, 19, 23
29   2
30   7, 160541, 94727075783
31   7, 79, 6451, 2806861
32   5, 1093, 3511
33   2^5, 233, 47441, 9639595369
34   46145917691
35   3, 1613, 3571
36   66161, 534851, 3152573
37   2, 3, 77867, 76407520781
38   17, 127
39   8039
40   11, 17, 307, 66431
41   2^3, 29, 1025273, 138200401
42   23^3, 719867822369
43   5, 103
44   3, 229, 5851
45   2, 1283, 131759, 157635607
46   3, 829
47
48   7, 257
49   2^4, 5, 491531
50   7
51   5, 41
52   461, 1228488439
53   2, 3^3, 47, 59, 97
54   19, 1949
55   3^3, 30109, 7278001, 27207490529, 902060958301
56   647, 7079771, 115755260963
57   2^3, 5^3, 47699, 86197
58   131, 42250279
59   2777
60   29, 9566295763
61   2
62   3, 19, 127, 1291
63   23, 29, 36713, 401771
64   3, 1093, 3511
65   2^6, 17, 163
66   89351671, 588024812497
67   7, 47, 268573
68   5^3, 7, 19, 113^3, 2741
69   2, 19, 223, 631, 2503037
70   13, 142963
71   3, 47, 331
72
73   2^3, 3
74   5
75   17, 43, 347, 31247
76   5, 37, 1109, 9241, 661049, 20724663983
77   2, 32687
78   43, 151, 181, 1163, 56149, 4229335793
79   7, 263, 3037, 1012573, 60312841
80   3^4, 7, 13, 6343
81   2^4, 11, 1006003
82   3^4, 5
83   4871, 13691, 315746063
84   163, 653, 20101, 663840051067
85   2, 11779
86   68239, 6232426549
87   1999, 48121, 604807523183
88   2535619637
89   2^3, 3, 13
90   6590291053
91   3, 293
92   727, 383951, 12026117, 18768727, 1485161969
93   2, 5, 509, 9221, 81551
94   11, 241, 32143, 463033
95   2137, 15061, 96185643031
96   109, 5437, 8329, 12925267, 103336004179
97   2^5, 7, 2914393, 76704103313
98   3, 28627, 61001527
99   5, 7, 13, 19, 83
100   3, 487, 56598313
101   2, 5, 1050139
102   7559, 11813, 139409857
103   24490789
104   313, 237977, 11950711691
105   2^3, 7669
106   79399, 672799
107   3^3, 5, 97, 613181
108   3761, 10271, 1296018233
109   2, 3^3, 20252173
110   17, 5381, 9431
111   131
112   11, 1037888513
113   2^4, 405697846751
114   9181
115   31, 2743780307
116   3, 7, 19, 47
117   2, 7, 31, 37, 182111
118   3, 5, 11, 23, 3152249, 10404887
119   1741
120   11, 653, 2074031, 124148023
121   2^3, 71
122   11, 2791, 16522744709
123   34849
124   5^3, 11^3, 22511
125   2, 3, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
126   5^3
127   3, 19, 907, 13778951
128   7, 1093, 3511
129   2^7, 7, 113
130   11, 23, 62351, 70439
131   17, 754480919
132   5
133   2, 5277179
134   3^3, 17, 406531, 73629977
135   23147324413
136   3^3, 5153, 87249620417
137   2^3, 29, 59, 6733, 18951271, 4483681903
138   97, 4889
139
140   14591
141   2, 192047, 42039743
142   143111, 12838109
143   3, 5, 67, 197, 1999, 12848347, 65451824959, 225413072431
144   2693, 123653
145   2^4, 3, 31
146   7, 13, 79, 22639
147   13, 79, 103, 283, 24203, 4058948753
148   7, 11, 41, 14879, 161141, 2805469039
149   2, 5, 29573, 121456243, 2283131621
150   13, 21199069, 85877201, 103053130679
151   5, 2251, 14107, 5288341, 15697215641
152   3, 22554863, 54504719
153   2^3, 8539, 5540099
154   3
155   17, 211691, 309131, 6282943, 1196757167
156   347, 1297
157   2, 5, 122327, 4242923, 5857727461, 275318049829
158   17^3, 1031, 820279
159   367, 5356051
160   72669541193
161   2^5, 3^4, 11^3, 83, 137, 6188401, 93405163
162   607, 142946059423
163   3^4, 3898031
164   103, 107, 1493
165   2, 7, 211, 25102757, 11699444843
166   7, 877, 61819
167   64661497
168   5, 13, 739, 27427
169   2^3, 863, 1747591
170   3, 13, 23, 1163, 170613512651
171   269, 1803383
172   3, 47, 91303
173   2, 3079, 56087
174   5, 113, 199, 379
175   397, 487, 2199439603
176   5, 191, 5611481459
177   2^4, 7, 23, 42397, 123681208241
178   7, 1213, 67751, 296036443
179   3, 17, 35059, 126443
180   269, 5737
181   2, 3, 101
182   5^4, 211, 8934721
183   1231487
184   89, 661, 1187, 2203, 6043
185   2^3, 12577
186
187
188   3^3, 13
189   2, 1847, 6199
190   3^3, 29, 236981, 382351, 753743
191   13, 379133
192   13, 4049, 302279, 465435739
193   2^6, 5^3, 4877
194   439, 8338826207
195   7, 23, 13421, 3132631
196   29, 353, 7596952219
197   2, 3, 7, 653, 6237773
198   147647, 545639981
199   3, 5, 77263, 1843757
200
201   2^3, 5, 86722091
202   11, 47
203
204   5431, 142061, 1070347, 15028133
205   2, 367
206   3, 151, 1201, 2087
207   5, 41, 269, 809, 26240063
208   3, 947
209   2^4, 229, 2203, 511654217, 6802750783
210   43, 449
211   279311
212   1960573
213   2, 313, 1877
214   7, 17
215   3^3, 7, 11, 3327697, 502095466241
216   66161, 534851, 3152573
217   2^3, 3^3, 127447, 8468323
218   5, 163
219   7891033151
220   3581, 9619, 79867, 717967, 13477271, 32376410527, 34816906301
221   2, 29, 1669, 15233, 103361803, 406833359, 455317162337, 1180548631831
222
223   71, 349
224   3, 5, 17, 130677149
225   2^5, 29131, 119327070011
226   3, 5, 7, 97, 157
227   7, 40277
228   1361, 182353
229   2, 31
230   47, 384533887
231
232   5, 30851, 838667, 2250741293
233   2^3, 3, 11, 157, 86735239
234   19, 46441, 817766595407
235   3, 31, 118022689
236   29, 229
237   2
238   977, 8609, 4997219, 863039377, 64904716187
239   11, 13^4, 74047, 212855197, 361552687, 12502228667
240   227797
241   2^4, 11, 523, 1163, 35407
242   3^5, 373, 2386301, 2434609, 167962252433
243   5, 11, 1006003
244   3^5, 7, 16906680221
245   2, 11, 19, 229, 218173429
246   7, 877441867
247   5081, 105527103139
248   67, 101, 663893, 502608046943, 668677573093
249   2^3, 5^3, 13, 17
250   131, 509
251   3, 5^3, 11, 17, 421, 395696461
252   211, 997, 3613, 394213789
253   2, 3, 167, 90901, 2905031
254   89, 2269, 18042251, 32984389
255   23, 463
256   1093, 3511
257   2^8, 5, 359, 49559, 648258371
258   13, 17827
259   173
260   3, 83, 15263, 27827, 1749229, 6647339, 19148231, 3977977109
261   2, 43, 71, 257
262   3, 19, 571
263   7, 23, 251, 267541, 159838801
264   7, 61, 176051, 1446587
265   2^3, 10853
266   23, 163, 21401
267   29, 20411, 46926349
268   5, 13, 101
269   2, 3^3, 11, 83, 8779, 65684482177
270   29
271   3^3, 168629, 16774141, 235558417, 12145092821
272   3833, 14084849
273   2^4, 1086731
274   5, 23, 16631
275   7, 6323, 134513, 138641, 455471, 40196043527, 279203122727
276   5, 7, 43, 193
277   2, 1993, 243547988443
278   3, 1741
279   67, 97, 557142359
280   3, 47, 401
281   2^3, 3443059
282   5, 11, 13120561
283   46301
284   197, 1103, 9127, 16189, 1601147, 10723208449
285   2, 5791, 157907
286   71, 619, 15017, 1016689
287   3, 653, 967, 10322313269, 405469792823
288   17, 43, 36979, 376721, 670507
289   2^5, 3, 46021, 48947, 478225523351
290   17, 6659, 319811, 273354437
291   109, 16766930077
292   19, 43, 887, 941, 3239857
293   2, 5, 7, 19, 83
294   356135477, 592538407, 367003576739
295   7, 47^3, 26249
296   3^3, 971
297   2^3, 2480497, 2779225651
298   3^3, 59, 101, 181, 598421
299   5, 19, 59, 374893957
300   37, 59, 29599

2022-12-09, 07:24   #6
preda

"Mihai Preda"
Apr 2015

5·172 Posts

Quote:
 Originally Posted by sweety439 Code:  Basis Primzahlen^2 (P^2) 1 All primes^infinity 2 1093, 3511 3 11, 1006003 4 1093, 3511 ...
what is in that list?

2022-12-09, 09:25   #7
S485122

"Jacob"
Sep 2006
Brussels, Belgium

1,907 Posts

Quote:
 Originally Posted by sweety439 ... Since for all positive integer n we have 1^(p-1) == 1 mod p^n ...
Couldn't you simplify that :

"Since for any integers n and p, with 0<n and 1<p, we have 1 == 1 mod p^n"

or :

"Since 1=1"

or even drop that part altogether because it is always true.

2022-12-09, 10:05   #8
kruoli

"Oliver"
Sep 2017
Porta Westfalica, DE

24×83 Posts

Quote:
 Originally Posted by preda what is in that list?
It looks like numbers, where $$b^{p-1} \equiv 1 \mod p^2$$ where $$b$$ is "Basis" and $$p$$ is "Primzahlen".

2022-12-10, 06:03   #9
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

368210 Posts

Quote:
 Originally Posted by preda what is in that list?
They are the primes p such that b^(p-1) == 1 mod p^2 (called "generalized Wieferich primes base b"), if "p^3" or "p^4" are written, then this means b^(p-1) == 1 mod p^3 (or p^4), e.g. b = 18 has 7^3, and this means 18^(7-1) == 1 mod 7^3

This is the full list for bases 2 <= b <= 10000 (not include b = 1)
Attached Files
 full list of generalized Wieferich.txt (273.5 KB, 15 views)

Last fiddled with by sweety439 on 2022-12-10 at 06:04

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