20220306, 12:23  #1 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×7×263 Posts 
(conjectured or proved) finiterecord sequences
only list the baseindependent 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 {256=19, 3527=8, 602596=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^n1 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^n1 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^nk 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^n509203 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^nk 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^n509203 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 20220504 at 02:35 
20220306, 13:44  #2 
"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 22122513digit 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 20220306 at 13:59 
20220424, 20:29  #3 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
3682_{10} 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.

20220514, 13:36  #4  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·7·263 Posts 
Quote:
* infinity is > any finite number, this is more convenient (e.g. the smallest n such that (k^n1)/(k1) is prime, for k = 185, (k^n1)/(k1) 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^n1)/(k1) is prime for k = 185) * "the smallest n such that (k^n1)/(k1) is prime" is "the largest n such that (k^r1)/(k1) is not prime for all r<n", for the case that (k^n1)/(k1) 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^n1)/(k1) is prime}, a(n) = min(S(k)), then a(n) = the smallest n such that (k^n1)/(k1) 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 20221201 at 15:41 

20221208, 23:27  #5 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
7142_{8} Posts 
Another example of "infinity" in number theory:
Since for all positive integer n we have 1^(p1) == 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 
20221209, 07:24  #6 
"Mihai Preda"
Apr 2015
5·17^{2} Posts 

20221209, 09:25  #7 
"Jacob"
Sep 2006
Brussels, Belgium
1,907 Posts 

20221209, 10:05  #8 
"Oliver"
Sep 2017
Porta Westfalica, DE
2^{4}×83 Posts 

20221210, 06:03  #9 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
3682_{10} Posts 
They are the primes p such that b^(p1) == 1 mod p^2 (called "generalized Wieferich primes base b"), if "p^3" or "p^4" are written, then this means b^(p1) == 1 mod p^3 (or p^4), e.g. b = 18 has 7^3, and this means 18^(71) == 1 mod 7^3
This is the full list for bases 2 <= b <= 10000 (not include b = 1) Last fiddled with by sweety439 on 20221210 at 06:04 
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