Index Nr | Base Sequence Expression | Initials |
---|---|---|
Various Palindromic Sums | Length | |
Multigrade Palprimes Patterns Contribution by Carlos Rivera [ August 13, 1999 ] See also Carlos' Puzzle 65 on Multigrade Relations. | ||
2 | 10501 + 14741 + 15451 = 11411 + 12721 + 16561 105012 + 147412 + 154512 = 114112 + 127212 + 165612 |
|
1 | 181 + 727 + 757 = 353 + 383 + 929 1812 + 7272 + 7572 = 3532 + 3832 + 9292 |
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! | Next is an ingenuously beautiful trigrade equation using palindromes from Albert Beiler's book “Recreations in the Theory of Numbers” 13031 + 42024 + 53035 + 57075 + 68086 + 97079 = 330330 = 31013 + 24042 + 35053 + 75057 + 86068 + 79097 130312 + 420242 + 530352 + 570752 + 680862 + 970792 = 22066126024 = 310132 + 240422 + 350532 + 750572 + 860682 + 790972 130313 + 420243 + 530353 + 570753 + 680863 + 970793 = 1642056213257460 = 310133 + 240423 + 350533 + 750573 + 860683 + 790973 |
Trigrade from A. Beiler's book page 164 |
Sums of Squares of Consecutive Odd Numbers![]() Entry 1 by Kimberly Pellechi [ July 1, 2003 ] Entry 2 by Hugo Sánchez [ March 17, 1999 ] Entries 3 to 17 by Kimberly Pellechi [ July 25, 2003 ] | ||
17 | 42284612 + 42284632 + ... + 42284792 + 42284812 | KP |
196.679.636.976.691 | 15 | |
16 | 10111092 + 10111112 + ... + 10111272 + 10111292 | KP |
11.245.977.954.211 | 14 | |
15 | 4246692 + 4246712 + ... + 4246872 + 4246892 | KP |
1.983.874.783.891 | 13 | |
14 | 1249512 + 1249532 + ... + 1249692 + 1249712 | KP |
171.767.767.171 | 12 | |
13 | 1190912 + 1190932 + ... + 1191092 + 1191112 | KP |
156.035.530.651 | 12 | |
12 | 1011092 + 1011112 + ... + 1011272 + 1011292 | KP |
112.475.574.211 | 12 | |
11 | 424192 + 424212 + ... + 424372 + 424392 | KP |
19.802.420.891 | 11 | |
10 | 131612 + 131632 + ... + 131792 + 131812 | KP |
1.908.228.091 | 10 | |
9 | 110812 + 110832 + ... + 110992 + 111012 | KP |
1.353.113.531 | 10 | |
8 | 101212 + 101232 + ... + 101392 + 101412 | KP |
1.129.009.211 | 10 | |
7 | 101092 + 101112 + ... + 101272 + 101292 | KP |
1.126.336.211 | 10 | |
6 | 11912 + 11932 + 11952 + 11972 + 11992 + 12012 + 12032 + 12052 + 12072 + 12092 + 12112 | KP |
15.866.851 | 8 | |
5 | 10092 + 10112 + 10132 + 10152 + 10172 + 10192 + 10212 + 10232 + 10252 + 10272 + 10292 | KP |
11.422.411 | 8 | |
4 | 992 + 1012 + 1032 + 1052 + 1072 + 1092 + 1112 + 1132 + 1152 + 1172 + 1192 | KP |
131.131 | 6 | |
3 | 292 + 312 + 332 + 352 + 372 + 392 + 412 + 432 + 452 + 472 + 492 | KP |
17.171 | 5 | |
2 | 212 + 232 + 252 + 272 + 292 + 312 + 332 + 352 + 372 + 392 + 412 | HS |
11.011 | 5 | |
1 | 12 + 32 + 52 + 72 + 92 + 112 + 132 + 152 + 172 + 192 + 212 | KP |
1.771 | 4 | |
Sums of Squares of Consecutive Odd Numbers![]() Entry 1 by Hugo Sánchez [ March 17, 1999 ] Entries 2 & 3 by Kimberly Pellechi [ July 25, 2003 ] | ||
3 | 790272 + 790292 + ... + 790392 + 790412 | KP |
56.218.781.265 | 11 | |
2 | 34652 + 34672 + ... + 34792 + 34812 | KP |
108.555.801 | 9 | |
1 | 12 + 32 + 52 + 72 + 92 + 112 + 132 + 152 + 172 | HS |
969 | 3 | |
Sums of Squares of Consecutive Odd Numbers![]() Contribution by Kimberly Pellechi [ July 1 & 25, 2003 ] | ||
2 | 85672 + 85692 + 85712 + 85732 + 85752 + 85772 + 85792 | KP |
514.474.415 | 9 | |
1 | 52 + 72 + 92 + 112 + 132 + 152 + 172 | KP |
959 | 3 | |
Sums of Squares of Consecutive Odd Numbers![]() Contribution by Kimberly Pellechi [ July 14, 2003 ] | ||
3 | 107892 + 107912 + 107932 + 107952 + 107972 | KP |
582.444.285 | 9 | |
2 | 103952 + 103972 + 103992 + 104012 + 104032 | KP |
540.696.045 | 9 | |
1 | 3312 + 3332 + 3352 + 3372 + 3392 | KP |
561.165 | 6 | |
Sums of Squares of Consecutive Odd Numbers![]() Entries 3 & 4 by Kimberly Pellechi [ July 1, 2003 ] Extended by Patrick De Geest [ July 7, 2003 ] | ||
19 | 71.818.1892 + 71.818.1912 + 71.818.1932 | PDG |
15.473.557.675.537.451 | 17 | |
18 | 67.315.7192 + 67.315.7212 + 67.315.7232 | PDG |
13.594.218.881.249.531 | 17 | |
17 | 41.386.1352 + 41.386.1372 + 41.386.1392 | PDG |
5.138.437.007.348.315 | 16 | |
16 | 8.008.8172 + 8.008.8192 + 8.008.8212 | PDG |
192.423.545.324.291 | 15 | |
15 | 7.181.8072 + 7.181.8092 + 7.181.8112 | PDG |
154.735.141.537.451 | 15 | |
14 | 718.1892 + 718.1912 + 718.1932 | PDG |
1.547.394.937.451 | 13 | |
13 | 413.8612 + 413.8632 + 413.8652 | PDG |
513.847.748.315 | 12 | |
12 | 113.5132 + 113.5152 + 113.5172 | PDG |
38.656.965.683 | 11 | |
11 | 11.3732 + 11.3752 + 11.3772 | PDG |
388.171.883 | 9 | |
10 | 6.7192 + 6.7212 + 6.7232 | PDG |
135.515.531 | 9 | |
9 | 4.1352 + 4.1372 + 4.1392 | PDG |
51.344.315 | 8 | |
8 | 1.1332 + 1.1352 + 1.1372 | PDG |
3.864.683 | 7 | |
7 | 7592 + 7612 + 7632 | PDG |
1.737.371 | 7 | |
6 | 7072 + 7092 + 7112 | PDG |
1.508.051 | 7 | |
5 | 1972 + 1992 + 2012 | PDG |
118.811 | 6 | |
4 | 792 + 812 + 832 | KP |
19.691 | 5 | |
3 | 412 + 432 + 452 | KP |
5.555 | 4 | |
2 | 192 + 212 + 232 | PDG |
1.331 | 4 | |
1 | 112 + 132 + 152 | PDG |
515 | 3 | |
Sums of Squares of Consecutive Even Numbers![]() Entries found by Patrick De Geest [ July 13, 2003 ] | ||
3 | 16.403.4682 + 16.403.4702 + 16.403.4722 | PDG |
807.221.484.122.708 | 15 | |
2 | 482.1842 + 482.1862 + 482.1882 | PDG |
697.510.015.796 | 12 | |
1 | 145.4242 + 145.4262 + 145.4282 | PDG |
63.446.164.436 | 11 | |
Diophantine Equation - 3rd & 4th powers Sources : Puzzle 47 & Puzzle 48 copied from Carlos Rivera's PP&P site. | ||
2 | 693 + 4473 + 8933 | 3 terms |
9293 | 3 | |
1 | 304 + 1204 + 2724 + 3154 | 4 terms |
3534 | 3 | |
Diophantine Equations - 2nd powers From Hugo Sánchez [ May 3, 1999 ] | ||
3 | 110112 + 220222 + 330332 + 660662 = 2 x 550552 | 4 terms |
2 x 550552 = 330332 + 440442 + 550552 | 3 terms | |
2 | 2122 + 3432 + 4242 + 9792 = 1300810 | 4 terms |
1300810 = 5552 + 6362 + 7672 | 3 terms | |
1 | 222 + 332 + 442 + 992 = 13310 | 4 terms |
13310 = 552 + 662 + 772 | 3 terms | |
Palindromic Pattern from Sums of Consecutives By Carlos Rivera [ Feb 27, 1999 ] | ||
1 | S(2 + 3 + 4) = 9 S(2 + 3 + ... + 44) = 989 S(2 + 3 + ... + 444) = 98789 S(2 + 3 + ... + 4444) = 9876789 S(2 + 3 + ... + 44444) = 987656789 S(2 + 3 + ... + 444444) = 98765456789 S(2 + 3 + ... + 4444444) = 9876543456789 S(2 + 3 + ... + 44444444) = 987654323456789 S(2 + 3 + ... + 444444444) = 98765432123456789 S(2 + 3 + ... + 4444444444) = 9876543210123456789 | pattern is finite ! |
Thank you Carlos for this beautiful construction. | ||
Palindromic Pattern from Sums of Squared Palindromes By Hugo Sánchez [ May 3, 1999 ] | ||
1 | 112 + 222 + ... + 662 = 91 x 112 = 11011 1112 + 2222 + ... + 6662 = 91 x 1112 = 1121211 11112 + 22222 + ... + 66662 = 91 x 11112 = 112323211 111112 + 222222 + ... + 666662 = 91 x 111112 = 11234343211 1111112 + 2222222 + ... + 6666662 = 91 x 1111112 = 1123454543211 11111112 + 22222222 + ... + 66666662 = 91 x 11111112 = 112345656543211 111111112 + 222222222 + ... + 666666662 = 91 x 111111112 = 11234567676543211 1111111112 + 2222222222 + ... + 6666666662 = 91 x 1111111112 = 1123456787876543211 11111111112 + 22222222222 + ... + 66666666662 = 91 x 11111111112 = 112345678989876543211 | pattern is finite ! |
Thanks Hugo for this beautiful pattern. Note that 91 is in fact a pseudopalindrome 1n1 See my palindromic squares page for more information | ||
Palindromic Sum of Powers from Consecutives By Carlos Rivera [ Feb 27, 1999 ] | ||
4 | 15 + 25 + 35 + 45 + 55 + 65 + 75 + 85 + 95 + 105 + 115 + 125 + 135 | 13 terms |
1.002.001 1002001 = 10012 = 72 x 112 x 132 | 7 | |
3 | 15 + 25 | 2 terms |
33 | 2 | |
2 | 14 + 24 + 34 + 44 + 54 | 5 terms |
979 | 3 | |
1 | 12 + 22 + 32 +... ...+ 1802 + 1812 For sum of squares with max 5 terms see Sum of Squares. | 181 terms |
1.992.991 | 7 | |
Palindromes from Consecutive Primes 2 to 23 and the Nine Digits Anagrams By definition the palindromes are always composite. By Carlos Rivera [ Feb 11, 1999 ] | ||
8 | 28 + 39 + 52 + 74 + 116 + 131 + 177 + 193 + 235 | 9 terms |
418.575.814 | 9 | |
7 | 23 + 36 + 59 + 75 + 118 + 137 + 174 + 192 + 231 | 9 terms |
279.161.972 | 9 | |
6 | 24 + 37 + 59 + 76 + 118 + 135 + 173 + 191 + 232 | 9 terms |
216.808.612 | 9 | |
5 | 27 + 38 + 51 + 79 + 112 + 134 + 175 + 196 + 233 | 9 terms |
88.866.888 | 8 | |
4 | 26 + 39 + 58 + 74 + 111 + 137 + 175 + 193 + 232 | 9 terms |
64.588.546 | 8 | |
3 | 29 + 38 + 56 + 73 + 117 + 132 + 171 + 194 + 235 | 9 terms |
26.077.062 | 8 | |
2 | 28 + 37 + 59 + 76 + 113 + 134 + 171 + 195 + 232 | 9 terms |
4.579.754 | 7 | |
1 | 28 + 39 + 57 + 71 + 116 + 134 + 173 + 195 + 232 | 9 terms |
4.379.734 | 7 | |
Palindromes from Consecutive Primes 3 to 29 and the Nine Digits Anagrams The palindromes have a chance to be prime. By Carlos Rivera [ Feb 11, 1999 ] | ||
4 | 39 + 58 + 74 + 111 + 133 + 176 + 192 + 237 + 295 | 9 terms |
3.449.889.443 | 10 | |
3 | 37 + 59 + 78 + 113 + 134 + 171 + 195 + 236 + 292 | 9 terms |
158.262.851 | 9 | |
2 | 34 + 58 + 79 + 112 + 137 + 176 + 195 + 231 + 293 | 9 terms |
130.131.031 | 9 | |
1 | 39 + 58 + 77 + 116 + 135 + 174 + 193 + 232 + 291 Note that the primes and the 9-digit anagram exponents are well ordered but in opposite direction ! See also WONplate 55 | 9 terms |
3.467.643 | 7 | |
A record palindrome using startprime 29, ninedigital 792436518 and base 2 By PDG [ Jun 10, 2022 ] | ||
1 | 297 + 319 + 372 + 414 + 433 + 476 + 535 + 591 + 618 | 9 terms |
218.175.385.351.779 (base 10) 110001100110110111101010010101111011011001100011 (base 2) | 15 48 | |
A record palindrome using consecutive Fibonacci terms and ninedigital 947862153 By Alexandru Petrescu [ Jun 17, 2022 ] | ||
1 | 39 + 54 + 87 + 138 + 216 + 342 + 551 + 895 + 1443 | 9 terms |
6.490.660.946 | 10 |
Hugo Sánchez (email) a 'profesor de Educación Media que cultiva la Matemática Recreativa'
from Caracas, Venezuela found some interesting sequences- go to topic 9 and topic 11.
Carlos Rivera (email) found among others this beautiful pattern- go to topic 10, topic 12, topic 13 and topic 14.
Kimberly Pellechi (email) found many 'palindromic sums of the squares of the consecutive odd numbers'
or Pellechi Palindromes for short- go to topic.
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