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Palindromes and their Prime Factors
Assignment 3
rood Palprim5.htm rood


Palindromes and their Prime Factors

1. Smallest palindromes with exactly n palindromic prime factors (cwm)
2. Smallest palindromes with exactly n prime factors (cwm)
3. Smallest palindromes with exactly n distinct prime factors
4. Smallest palindromes with exactly n distinct palindromic prime factors
5. Palindromes divided by the palindromic sum of their prime factors (cwm) is a palindrome
6. Palindromes divisible by the palindromic sum of their prime factors(cwm)
7. Palindromes divisible by the palindromic sum of their exclusive palindromic prime factors


The Tabels

1. Smallest palindromes with exactly n palindromic prime factors
    (counted with multiplicity).
Report extensions also to Sloane's "Online Encyclopedia".
Refer to sequence A046385

npalindromepalindromic prime factors
01-
122
242 * 2
382 * 2 * 2
4882 * 2 * 2 * 11
52522 * 2 * 3 * 3 * 7
627722 * 2 * 3 * 3 * 7 * 11
7827282 * 2 * 2 * 3 * 3 * 3 * 383
821122 * 2 * 2 * 2 * 2 * 2 * 3 * 11
942242 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
1084482 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
112369896322 * 2 * 2 * 2 * 2 * 2 * 3 * 11 * 11 * 101 * 101
12483842 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 * 7
1329777922 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 727  
14??
15??

2. Smallest palindromes with exactly n prime factors
    (counted with multiplicity).
Refer to sequence A076886 [ November 25, 2002 ]
by Shyam Sunder Gupta (extended by Robert G. Wilson v).

npalindromeprime factors
01-
122
242 * 2
382 * 2 * 2
4882 * 2 * 2 * 11
52522 * 2 * 3 * 3 * 7
627722 * 2 * 3 * 3 * 7 * 11
7278722 * 2 * 2 * 2 * 2 * 13 * 67
821122 * 2 * 2 * 2 * 2 * 2 * 3 * 11
942242 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
1084482 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
11445442 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 29
12483842 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 * 7
1329777922 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 727  
14270110722 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 11 * 11 * 109
154055042 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 11
164091904212 * 33 * 37
17441606144213 * 3 * 7 * 17 * 151
18405909504212 * 32 * 7 * 112 * 13
19886898688216 * 3 * 13 * 347
20677707776216 * 33 * 383
214285005824219 * 11 * 743
22276486684672218 * 3 * 11 * 31 * 1031
2321128282112219 * 3 * 7 * 19 * 101
24633498894336217 * 3 * 74 * 11 * 61
252701312131072222 * 3 * 19 * 11299
268691508051968222 * 3 * 7 * 101 * 977

3. Smallest palindromes with exactly n distinct prime factors.
Report extensions also to Sloane's "Online Encyclopedia".
Refer to sequence A046399
From index nr 10 to 13 from Donovan Johnson
From index nr 14 to 35 from Giovanni Resta.
Entries copied dd. 13 dec 2020.

npalindromedistinct prime factors
01-
122
262 * 3
3662 * 3 * 11
48582 * 3 * 11 * 13
560062 * 3 * 7 * 11 * 13
62222222 * 3 * 7 * 11 * 13 * 37
7224444222 * 3 * 7 * 11 * 13 * 37 * 101
82448684422 * 3 * 7 * 13 * 17 * 23 * 31 * 37
964347743462 * 3 * 7 * 11 * 13 * 17 * 19 * 31 * 107
104380244208342 * 3 * 7 * 11 * 13 * 17 * 19 * 43 * 59 × 89
11501469559641053 * 5 * 7 * 11 * 13 * 17 * 19 * 37 * 43 * 67 * 97
1224159579975951422 * 3 * 7 * 11 * 13 * 17 * 23 * 29 * 31 * 37 * 157 * 197
134956771211217765942 * 3 * 7 * 11 * 13 * 17 * 23 * 31 * 43 * 71 * 73 * 137 * 223
14221816737557376181222 * 3 * 7 * 11 * 13 * 17 * 19 * 23 * 31 * 37 * 83 * 101 * 149 * 347
1587899411649946114998782 * 3 * 7 * 11 * 13 * 17 * 19 * 23 * 37 * 41 * 79 * 83 * 173 * 239 * 479

4. Smallest palindromes with exactly n distinct palindromic prime factors.
Variation on the theme by Amarnath Murthy see A087331

npalindromedistinct palindromic prime factors
01-
122
262 * 3
3662 * 3 * 11
466662 * 3 * 11 * 101
5??
6??
7??

5. Palindromes divided by the palindromic sum of their prime factors
    (counted with multiplicity) is a palindrome.
Report extensions also to Sloane's "Online Encyclopedia".
Refer to sequence A046362
From index nr 5 to 12 from Donovan Johnson
From index nr 13 to 39 from Giovanni Resta.
Entries copied dd. 12 dec 2020.

npalindromeprime factorspalindromic sumpalindromic quotient
142 * 241
213454313 * 17 * 23 * 31 * 37 (all distinct)11112121
3123333213 * 3 * 7 * 11 * 13 * 37 * 37111111111
414256652413 * 3 * 7 * 7 * 11 * 13 * 13 * 37 * 4714110111101
5120262552620213 * 3 * 7 * 11 * 13 * 113 * 127 * 167 * 557100112014241021
6356866666686537 * 11 * 11 * 11 * 13 * 31 * 101 * 139 * 677100135651015653
7401819559181042 * 2 * 2 * 7 * 7 * 7 * 7 * 11 * 13 * 101 * 241 * 601100140141814104
81284494449448217 * 11 * 11 * 13 * 19 * 31 * 167 * 233 * 5091001128321123821
914642678762464173 * 137 * 1433 * 1487 * 6871 (all distinct)1000114641214641
106517511011571562 * 2 * 7 * 11 * 11 * 13 * 17 * 43 * 197 * 211 * 4871001651100001156
11173457388375437111 * 31 * 59 * 73 * 137 * 991 * 8699 (all distinct)10001173440044371
1223134212212431322 * 2 * 3 * 3 * 7 * 11 * 13 * 17 * 29 * 37 * 47 * 103 * 72710012311110111132
1368588742247885862 * 3 * 7 * 11 * 11 * 11 * 13 * 41 * 47 * 71 * 101 * 68310016852022202586
1498898820028898897 * 11 * 13 * 71 * 101 * 149 * 149 * 229 * 27110019880002000889
15124466867686644213 * 3 * 3 * 3 * 7 * 7 * 13 * 13 * 37 * 37 * 37 * 37 * 9901101011232223222321
16232652530352562322 * 2 * 2 * 7 * 7 * 11 * 11 * 13 * 13 * 29 * 29 * 47 * 101 * 727100123242011024232
17301533380833351037 * 13 * 73 * 137 * 2663 * 3119 * 3989 (all distinct)100013015032305103
18353036310136303533 * 11 * 11 * 59 * 73 * 101 * 137 * 173 * 9433100013530010100353
19511693488843961155 * 31 * 73 * 137 * 2713 * 3041 * 4001 (all distinct)100015116423246115
20748989352539898473 * 3 * 3 * 7 * 11 * 11 * 13 * 37 * 293 * 4243 * 5477101017415002005147
211132749699694723113 * 7 * 7 * 11 * 13 * 13 * 19 * 37 * 79 * 821 * 90911010111214233241211
2211375437777345731111 * 73 * 79 * 137 * 659 * 3761 * 5281 (all distinct)1000111374300347311
232111269955996211122 * 2 * 2 * 3 * 3 * 11 * 37 * 73 * 89 * 137 * 929 * 87131000121110588501112
242204717766771740222 * 3 * 3 * 7 * 7 * 11 * 13 * 13 * 13 * 17 * 41 * 101 * 349 * 4211001220251525152022
252766287711778266722 * 2 * 2 * 2 * 11 * 17 * 73 * 137 * 449 * 3623 * 56831000127660111106672
263644656266265644633 * 7 * 11 * 13 * 37 * 43 * 103 * 103 * 131 * 131 * 4191001364101525101463
274122688899888622142 * 3 * 7 * 11 * 73 * 103 * 137 * 151 * 647 * 8867 (all distinct)1000141222766722214
284227527644672572242 * 2 * 2 * 3 * 7 * 11 * 13 * 19 * 19 * 37 * 83 * 149 * 307 * 3471001422330434033224
2915589847434748985513 * 7 * 19 * 41 * 79 * 271 * 601 * 797 * 9293 (all distinct)11111140310030013041
3020547088808880745022 * 3 * 7 * 13 * 37 * 73 * 137 * 1069 * 1291 * 7369 (all distinct)10001205450343054502
3122464677767776464222 * 3 * 7 * 7 * 11 * 11 * 13 * 23 * 41 * 211 * 3313 * 736911011204020323020402
3223848836121638848322 * 2 * 2 * 2 * 2 * 7 * 7 * 11 * 11 * 13 * 23 * 43 * 89 * 97 * 269 * 42110012382501111052832
3330653766484667356037 * 11 * 11 * 13 * 53 * 79 * 101 * 103 * 113 * 163 * 34710013062314334132603
3434037126444621730437 * 11 * 11 * 13 * 41 * 89 * 97 * 107 * 151 * 191 * 28310013400312332130043
3553536669262966635353 * 5 * 11 * 11 * 31 * 73 * 73 * 101 * 137 * 137 * 941910001535313161313535
3656361563626365163653 * 5 * 11 * 11 * 11 * 29 * 9091 * 13921 * 7691910000156361000016365
3757268213202312862753 * 3 * 5 * 5 * 7 * 11 * 11 * 13 * 37 * 79 * 101 * 787 * 994911011520100020001025
3875219796949697912573 * 19 * 37 * 53 * 73 * 137 * 419 * 2311 * 6949 (all distinct)10001752122757221257
3986542887444788245682 * 2 * 2 * 3 * 3 * 3 * 11 * 11 * 47 * 9091 * 9533 * 8129310000186542022024568

6. Palindromes divisible by the palindromic sum of their prime factors
    (counted with multiplicity).
Report extensions also to Sloane's "Online Encyclopedia".
Refer to sequence A046360
From index nr 3 on the terms are taken from the OEIS sequence dd. 13 dec 2020.

npalindromepalindromic prime factorspalindromic sumquotient
142 * 241
254453 * 3 * 5 * 11 * 1133165
32344322 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 11 * 37663552
45855853 * 3 * 5 * 7 * 11 * 13 * 135510647
58888882 * 2 * 2 * 3 * 7 * 11 * 13 * 377711544
69511593 * 11 * 19 * 37 * 41 (all distinct)1118569
79999993 * 3 * 3 * 7 * 11 * 13 * 377712987
813454313 * 17 * 23 * 31 * 37 (all distinct)11112121
925545522 * 2 * 2 * 7 * 11 * 11 * 13 * 297733176
1046202642 * 2 * 2 * 3 * 11 * 11 * 37 * 4311141624
1158424853 * 3 * 5 * 11 * 11 * 29 * 379959015
1261515162 * 2 * 7 * 19 * 31 * 37343414174
1397040793 * 3 * 7 * 11 * 11 * 19 * 6712180199
14123333213 * 3 * 7 * 11 * 13 * 37 * 37111111111
15401991042 * 2 * 2 * 2 * 2 * 2 * 11 * 11 * 29 * 179242166112
16422662242 * 2 * 2 * 2 * 7 * 7 * 11 * 13 * 13 * 2988480298
17426666242 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 7 * 11 * 13 * 3788484848
18440880442 * 2 * 7 * 7 * 11 * 11 * 11 * 13 * 1377572572
19463553642 * 2 * 3 * 3 * 11 * 19 * 61 * 101202229482
20635995362 * 2 * 2 * 2 * 7 * 11 * 11 * 13 * 19 * 1988722722
21636336362 * 2 * 3 * 3 * 11 * 37 * 43 * 101202315018
224208380242 * 2 * 2 * 29 * 61 * 131 * 227454926956
234248484242 * 2 * 2 * 7 * 7 * 11 * 11 * 13 * 13 * 531213511144
246319491362 * 2 * 2 * 2 * 3 * 7 * 13 * 13 * 31 * 3594341456104
256493639462 * 7 * 11 * 83 * 101 * 503 (all distinct)707918478

7. Palindromes divisible by the palindromic sum of their exclusive palindromic prime factors.
    Nonpalindromic prime factors may not occur.
    (counted with multiplicity).

npalindromepalindromic prime factorspalindromic sumquotient
142 * 241
254453 * 3 * 5 * 11 * 1133165
3????
4????







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