[seqfan] Re: A027763 and related sequences regarding smallest k such that b^^n is not congruent to b^^(n-1) mod k, for constant b.
Wayne VanWeerthuizen
waynemv at gmail.com
Wed Aug 27 18:27:08 CEST 2014
First, here's an errata to an earlier post of mine in this thread. The line near the end should have read:
b=22: 2, [2^2], 5, [5^2], 83, 167, 503, 3019, 14087, 84523, 482957, 1449167, 4782969, ...
The error is that I had transposed the terms when manually replacing 25 with [5^2].
Moving on, per the previous discussion I have a program that for a given b, searches for "the smallest k such that b^^n is not congruent to b^^(n-1) mod k, where b^^n denotes the power tower b^b^...^b (in which b appears n times)." I've recently used the program to extend A027763 and to generate sequences A246491 through A246497. I uploaded the source code as a linked file in A246491.
During its searches it has found some interesting sets of numerical coincidences. These coincidences don't occur in every case, but do occur often enough to be interesting, I think. They may provide a clue to what makes these sequences behave as they do.
One set of coincidences is:
1. The lowest k, such that 5^^ 9 <> 23^^ 10 (mod k) is 59049
2. The lowest k, such that 5^^ 10 <> 23^^ 11 (mod k) is 177147
3. The lowest k, such that 5^^ 11 <> 23^^ 12 (mod k) is 531441
4. The lowest k, such that 5^^ 12 <> 23^^ 13 (mod k) is 1594323
5. The lowest k, such that 5^^ 13 <> 23^^ 14 (mod k) is 4782969
6. 5^^10 mod 59049 = 31889
7. 5^^11 mod 177147 = 31889
8. 5^^11 mod 531441 = 31889
9. 5^^12 mod 531441 = 386183
10. 5^^13 mod 1594323 = 386183
11. 5^^13 mod 4782969 = 386183
Another set is:
1. The lowest k, such that 23^^ 8 <> 23^^ 9 (mod k) is 19683.
2. The lowest k, such that 23^^ 9 <> 23^^10 (mod k) is 59049.
3. The lowest k, such that 23^^10 <> 23^^11 (mod k) is 177147.
4. The lowest k, such that 23^^11 <> 23^^12 (mod k) is 531441.
5. The lowest k, such that 23^^12 <> 23^^13 (mod k) is 1594323.
6. 23^^ 9 mod 19683 = 8597.
7. 23^^10 mod 59049 = 8597.
8. 23^^11 mod 177147 = 8597.
9. 23^^11 mod 531441 = 8597.
10. 23^^12 mod 531441 = 362891.
11. 23^^13 mod 1594323 = 362891.
I notice they only seem to occur when k is a power of three, and not when k is prime. But they don't always occur, even when k is a power of three. Does anyone have an explanation for why these coincidences occur some of the time, yet not all the time, with these sequences?
More of these coincidences can be seen in this raw data, compiled from multiple runs of my program:
Legend:
value is the lowest k that was found to fit the criteria per that base and index.
(A!=B) means:
A = (base^^index) mod value
B = (base^^index-1) mod value
with A unequal to B
base:2 index: 1 value: 2 (0!=1)
base:2 index: 2 value: 3 (1!=2)
base:2 index: 3 value: 5 (1!=4)
base:2 index: 4 value: 11 (9!=5)
base:2 index: 5 value: 23 (6!=9)
base:2 index: 6 value: 47 (17!=42)
base:2 index: 7 value: 283 (240!=168)
base:2 index: 8 value: 719 (507!=596)
base:2 index: 9 value: 1439 (95!=623)
base:2 index:10 value: 2879 (1025!=1043)
base:2 index:11 value: 34549 (3972!=1794)
base:2 index:12 value: 138197 (136820!=50988)
base:2 index:13 value: 531441 (500857!=323710)
base:2 index:14 value: 1594323 (500857!=1032298)
base:2 index:15 value: 4782969 (3689503!=2095180)
base:2 index:16 value: 14348907 (8472472!=13255441)
base:2 index:17 value: 43046721 (22821379!=8472472)
base:2 index:18 value: 86093443 (76182566!=34580636)
base:2 index:19 value: 344373773 (67084855!=210903255)
base:3 index: 1 value: 3 (0!=1)
base:3 index: 2 value: 5 (2!=3)
base:3 index: 3 value: 11 (9!=5)
base:3 index: 4 value: 23 (18!=13)
base:3 index: 5 value: 47 (6!=36)
base:3 index: 6 value: 283 (120!=279)
base:3 index: 7 value: 719 (573!=434)
base:3 index: 8 value: 1439 (1147!=992)
base:3 index: 9 value: 2879 (245!=1908)
base:3 index:10 value: 34549 (16748!=23969)
base:3 index:11 value: 138197 (107663!=110862)
base:3 index:12 value: 1266767 (225095!=1186711)
base:3 index:13 value: 14920303 (10714748!=12391602)
base:3 index:14 value: 36449279 (29667647!=17364648)
base:3 index:15 value: 377982107 (47526718!=306140320)
base:4 index: 1 value: 2 (0!=1)
base:4 index: 2 value: 5 (1!=4)
base:4 index: 3 value: 11 (4!=3)
base:4 index: 4 value: 23 (3!=18)
base:4 index: 5 value: 47 (17!=14)
base:4 index: 6 value: 283 (151!=51)
base:4 index: 7 value: 719 (676!=646)
base:4 index: 8 value: 1439 (775!=307)
base:4 index: 9 value: 2879 (1572!=2628)
base:4 index:10 value: 34549 (27005!=33286)
base:4 index:11 value: 138197 (129797!=124716)
base:4 index:12 value: 1266767 (94109!=851414)
base:4 index:13 value: 4782969 (2391484!=797161)
base:4 index:14 value: 14348907 (7174453!=2391484)
base:5 index: 1 value: 3 (2!=1)
base:5 index: 2 value: 7 (3!=5)
base:5 index: 3 value: 19 (6!=9)
base:5 index: 4 value: 47 (23!=44)
base:5 index: 5 value: 243 (56!=218)
base:5 index: 6 value: 719 (555!=227)
base:5 index: 7 value: 1439 (561!=376)
base:5 index: 8 value: 2879 (2417!=1655)
base:5 index: 9 value: 19683 (12206!=5645)
base:5 index:10 value: 59049 (31889!=51572)
base:5 index:11 value: 177147 (31889!=149987)
base:5 index:12 value: 531441 (386183!=31889)
base:5 index:13 value: 1594323 (386183!=1449065)
base:5 index:14 value: 4782969 (3574829!=386183)
base:6 index: 1 value: 2 (0!=1)
base:6 index: 2 value: 4 (0!=2)
base:6 index: 3 value: 13 (1!=12)
base:6 index: 4 value: 47 (36!=37)
base:6 index: 5 value: 107 (27!=40)
base:6 index: 6 value: 643 (225!=343)
base:6 index: 7 value: 1439 (1371!=685)
base:6 index: 8 value: 2879 (1876!=2184)
base:6 index: 9 value: 34549 (2795!=21378)
base:6 index:10 value: 138197 (78606!=28648)
base:6 index:11 value: 858239 (274700!=538573)
base:6 index:12 value: 2029439 (1154450!=1822767)
base:6 index:13 value: 36449279 (32351217!=23370726)
base:7 index: 1 value: 4 (3!=1)
base:7 index: 2 value: 5 (3!=2)
base:7 index: 3 value: 11 (2!=6)
base:7 index: 4 value: 23 (20!=19)
base:7 index: 5 value: 47 (37!=12)
base:7 index: 6 value: 283 (136!=94)
base:7 index: 7 value: 719 (334!=708)
base:7 index: 8 value: 1439 (1102!=1341)
base:7 index: 9 value: 2879 (1030!=688)
base:7 index:10 value: 34549 (3880!=17535)
base:7 index:11 value: 138197 (89122!=24924)
base:7 index:12 value: 1266767 (271352!=1133074)
base:7 index:13 value: 4782969 (3949711!=761065)
base:7 index:14 value: 14348907 (3949711!=13515649)
base:8 index: 1 value: 2 (0!=1)
base:8 index: 2 value: 3 (1!=2)
base:8 index: 3 value: 11 (3!=5)
base:8 index: 4 value: 23 (6!=16)
base:8 index: 5 value: 47 (25!=4)
base:8 index: 6 value: 283 (51!=134)
base:8 index: 7 value: 719 (681!=60)
base:8 index: 8 value: 1439 (957!=743)
base:8 index: 9 value: 2879 (1462!=426)
base:8 index:10 value: 34549 (877!=6450)
base:8 index:11 value: 138197 (75523!=94222)
base:8 index:12 value: 1266767 (944874!=1072918)
base:8 index:13 value: 14920303 (9934076!=3784062)
base:9 index: 1 value: 3 (0!=1)
base:9 index: 2 value: 7 (1!=2)
base:9 index: 3 value: 23 (8!=2)
base:9 index: 4 value: 47 (32!=34)
base:9 index: 5 value: 283 (253!=54)
base:9 index: 6 value: 719 (242!=416)
base:9 index: 7 value: 1439 (180!=1031)
base:9 index: 8 value: 2879 (2705!=327)
base:9 index: 9 value: 34549 (18374!=14979)
base:9 index:10 value: 138197 (34175!=53291)
base:9 index:11 value: 1266767 (509275!=1201399)
base:9 index:12 value: 14619833 (8600272!=8175299)
base:10 index: 1 value: 2 (0!=1)
base:10 index: 2 value: 4 (0!=2)
base:10 index: 3 value: 17 (1!=2)
base:10 index: 4 value: 47 (9!=24)
base:10 index: 5 value: 283 (227!=52)
base:10 index: 6 value: 719 (590!=245)
base:10 index: 7 value: 1439 (675!=481)
base:10 index: 8 value: 2879 (1313!=1525)
base:10 index: 9 value: 34549 (4963!=30690)
base:10 index:10 value: 138197 (37486!=32198)
base:10 index:11 value: 858239 (406071!=854561)
base:10 index:12 value: 3778253 (697054!=3247644)
base:11 index: 1 value: 3 (2!=1)
base:11 index: 2 value: 7 (2!=4)
base:11 index: 3 value: 17 (5!=12)
base:11 index: 4 value: 59 (39!=52)
base:11 index: 5 value: 163 (45!=2)
base:11 index: 6 value: 487 (377!=291)
base:11 index: 7 value: 1307 (852!=1035)
base:11 index: 8 value: 6561 (5342!=3155)
base:11 index: 9 value: 19683 (18464!=11903)
base:11 index:10 value: 59049 (57830!=38147)
base:11 index:11 value: 177147 (175928!=116879)
base:11 index:12 value: 531441 (353075!=175928)
base:11 index:13 value: 1594323 (1415957!=884516)
base:11 index:14 value: 4782969 (4604603!=3010280)
base:11 index:15 value: 14348907 (14170541!=9387572)
base:12 index: 1 value: 2 (0!=1)
base:12 index: 2 value: 5 (1!=2)
base:12 index: 3 value: 17 (1!=4)
base:12 index: 4 value: 83 (77!=51)
base:12 index: 5 value: 167 (7!=97)
base:12 index: 6 value: 503 (224!=177)
base:12 index: 7 value: 1439 (41!=1432)
base:12 index: 8 value: 2879 (1158!=2586)
base:12 index: 9 value: 34549 (3951!=1351)
base:12 index:10 value: 138197 (111377!=27643)
base:12 index:11 value: 858239 (786867!=583886)
base:12 index:12 value: 3778253 (2089972!=2584308)
base:13 index: 1 value: 5 (3!=1)
base:13 index: 2 value: 11 (8!=2)
base:13 index: 3 value: 23 (2!=8)
base:13 index: 4 value: 47 (19!=10)
base:13 index: 5 value: 283 (28!=9)
base:13 index: 6 value: 719 (482!=596)
base:13 index: 7 value: 1439 (1207!=901)
base:13 index: 8 value: 2879 (1527!=1380)
base:13 index: 9 value: 34549 (16013!=21767)
base:13 index:10 value: 138197 (12692!=56940)
base:13 index:11 value: 531441 (133996!=488290)
base:13 index:12 value: 1594323 (133996!=1196878)
base:13 index:13 value: 4782969 (3322642!=1728319)
base:13 index:14 value: 14348907 (12888580!=8105611)
base:14 index: 1 value: 2 (0!=1)
base:14 index: 2 value: 3 (1!=2)
base:14 index: 3 value: 9 (4!=7)
base:14 index: 4 value: 19 (17!=16)
base:14 index: 5 value: 47 (2!=42)
base:14 index: 6 value: 163 (15!=93)
base:14 index: 7 value: 487 (375!=285)
base:14 index: 8 value: 1307 (1283!=774)
base:14 index: 9 value: 2879 (79!=22)
base:14 index:10 value: 19683 (9148!=15709)
base:14 index:11 value: 39367 (21706!=31354)
base:14 index:12 value: 177147 (48514!=107563)
base:14 index:13 value: 531441 (225661!=402808)
base:14 index:14 value: 1594323 (757102!=1288543)
base:14 index:15 value: 4782969 (2351425!=3945748)
base:14 index:16 value: 14348907 (2351425!=7134394)
base:15 index: 1 value: 3 (0!=1)
base:15 index: 2 value: 9 (0!=6)
base:15 index: 3 value: 19 (18!=8)
base:15 index: 4 value: 47 (30!=33)
base:15 index: 5 value: 283 (253!=64)
base:15 index: 6 value: 719 (300!=187)
base:15 index: 7 value: 1439 (496!=1227)
base:15 index: 8 value: 2879 (728!=1253)
base:15 index: 9 value: 34549 (25446!=29334)
base:15 index:10 value: 138197 (40115!=114035)
base:15 index:11 value: 1266767 (1254007!=156367)
base:15 index:12 value: 8826479 (1086433!=7656158)
base:16 index: 1 value: 2 (0!=1)
base:16 index: 2 value: 17 (1!=16)
base:16 index: 3 value: 47 (2!=25)
base:16 index: 4 value: 163 (83!=87)
base:16 index: 5 value: 487 (235!=463)
base:16 index: 6 value: 1307 (986!=191)
base:16 index: 7 value: 2879 (2723!=76)
base:16 index: 8 value: 19683 (18331!=11770)
base:16 index: 9 value: 59049 (57697!=18331)
base:16 index:10 value: 177147 (175795!=116746)
base:16 index:11 value: 531441 (530089!=175795)
base:16 index:12 value: 1594323 (530089!=1592971)
base:16 index:13 value: 4782969 (530089!=2124412)
base:16 index:14 value: 14348907 (530089!=10096027)
base:17 index: 1 value: 3 (2!=1)
base:17 index: 2 value: 7 (5!=3)
base:17 index: 3 value: 23 (5!=11)
base:17 index: 4 value: 47 (34!=6)
base:17 index: 5 value: 283 (35!=98)
base:17 index: 6 value: 719 (680!=284)
base:17 index: 7 value: 1439 (1379!=434)
base:17 index: 8 value: 2879 (907!=1596)
base:17 index: 9 value: 34549 (31099!=31109)
base:17 index:10 value: 138197 (92896!=92682)
base:17 index:11 value: 1266767 (518655!=506874)
base:17 index:12 value: 14619833 (4739546!=6706535)
base:17 index:13 value: 36449279 (5076023!=5705429)
base:18 index: 1 value: 2 (0!=1)
base:18 index: 2 value: 4 (0!=2)
base:18 index: 3 value: 5 (1!=4)
base:18 index: 4 value: 11 (4!=3)
base:18 index: 5 value: 23 (4!=13)
base:18 index: 6 value: 47 (25!=2)
base:18 index: 7 value: 283 (66!=106)
base:18 index: 8 value: 719 (214!=374)
base:18 index: 9 value: 1439 (61!=171)
base:18 index:10 value: 2879 (257!=1331)
base:18 index:11 value: 34549 (453!=19910)
base:18 index:12 value: 138197 (1708!=83832)
base:18 index:13 value: 1266767 (47276!=283336)
base:19 index: 1 value: 4 (3!=1)
base:19 index: 2 value: 11 (7!=8)
base:19 index: 3 value: 23 (15!=14)
base:19 index: 4 value: 47 (39!=40)
base:19 index: 5 value: 257 (171!=86)
base:19 index: 6 value: 719 (88!=426)
base:19 index: 7 value: 1439 (192!=1219)
base:19 index: 8 value: 2879 (370!=2500)
base:19 index: 9 value: 34549 (30847!=25995)
base:19 index:10 value: 138197 (16704!=11311)
base:19 index:11 value: 1266767 (867947!=817836)
base:20 index: 1 value: 2 (0!=1)
base:20 index: 2 value: 3 (1!=2)
base:20 index: 3 value: 9 (7!=4)
base:20 index: 4 value: 27 (16!=25)
base:20 index: 5 value: 81 (43!=16)
base:20 index: 6 value: 163 (4!=69)
base:20 index: 7 value: 487 (442!=274)
base:20 index: 8 value: 1307 (587!=1006)
base:20 index: 9 value: 6561 (610!=4984)
base:20 index:10 value: 19683 (610!=7171)
base:20 index:11 value: 59049 (20293!=610)
base:20 index:12 value: 177147 (20293!=79342)
base:20 index:13 value: 531441 (197440!=20293)
base:21 index: 1 value: 3 (0!=1)
base:21 index: 2 value: 9 (0!=3)
base:21 index: 3 value: 19 (18!=8)
base:21 index: 4 value: 149 (78!=141)
base:21 index: 5 value: 383 (110!=279)
base:21 index: 6 value: 1439 (920!=1310)
base:21 index: 7 value: 2879 (232!=2044)
base:21 index: 8 value: 32633 (29169!=25297)
base:21 index: 9 value: 65267 (63480!=1431)
base:21 index:10 value: 913739 (598915!=126099)
base:21 index:11 value: 1827479 (1145043!=1817401)
base:21 index:12 value: 8826479 (2434485!=2741039)
base:22 index: 1 value: 2 (0!=1)
base:22 index: 2 value: 4 (0!=2)
base:22 index: 3 value: 5 (1!=4)
base:22 index: 4 value: 25 (21!=6)
base:22 index: 5 value: 83 (3!=41)
base:22 index: 6 value: 167 (127!=47)
base:22 index: 7 value: 503 (367!=344)
base:22 index: 8 value: 3019 (2761!=187)
base:22 index: 9 value: 14087 (11298!=3788)
base:22 index:10 value: 84523 (51151!=12713)
base:22 index:11 value: 482957 (294113!=421070)
base:22 index:12 value: 1449167 (558601!=31157)
base:22 index:13 value: 4782969 (1238818!=4427464)
base:22 index:14 value: 14348907 (1238818!=10804756)
base:23 index: 1 value: 3 (2!=1)
base:23 index: 2 value: 5 (2!=3)
base:23 index: 3 value: 19 (16!=17)
base:23 index: 4 value: 59 (38!=52)
base:23 index: 5 value: 167 (102!=45)
base:23 index: 6 value: 487 (176!=411)
base:23 index: 7 value: 1459 (576!=1125)
base:23 index: 8 value: 6561 (2036!=4223)
base:23 index: 9 value: 19683 (8597!=2036)
base:23 index:10 value: 59049 (8597!=28280)
base:23 index:11 value: 177147 (8597!=126695)
base:23 index:12 value: 531441 (362891!=8597)
base:23 index:13 value: 1594323 (362891!=1425773)
base:23 index:14 value: 4782969 (1957214!=3551537)
base:23 index:15 value: 14348907 (1957214!=11523152)
base:24 index: 1 value: 2 (0!=1)
base:24 index: 2 value: 5 (1!=4)
base:24 index: 3 value: 11 (9!=5)
base:24 index: 4 value: 59 (48!=46)
base:24 index: 5 value: 167 (116!=88)
base:24 index: 6 value: 359 (202!=48)
base:24 index: 7 value: 719 (472!=697)
base:24 index: 8 value: 1439 (667!=456)
base:24 index: 9 value: 2879 (2207!=1624)
base:24 index:10 value: 34549 (30140!=13981)
base:24 index:11 value: 138197 (45119!=74493)
base:24 index:12 value: 1266767 (1195197!=6934)
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