HOME plateWON | World!OfNumbers Palindromic Sums ofCubes of Consecutive Integers Sums of Squares Sums of Primes Sums of Powers Various Palindromic Sums

Introduction

Palindromic numbers are numbers which read the same from
left to right (forwards) as from the right to left (backwards)
Here are a few random examples : 535, 3773, 246191642

Palindromic Sums of Cubes of Consecutive Integers

General formulae ( A*x3 + B*x2 + C*x + D )

SOCU2 = 2*x^3 + 3*x^2 + 3*x + 1
SOCU3 = 3*x^3 + 9*x^2 + 15*x + 9
SOCU4 = 4*x^3 + 18*x^2 + 42*x + 36
SOCU5 = 5*x^3 + 30*x^2 + 90*x + 100
SOCU6 = 6*x^3 + 45*x^2 + 165*x + 225
SOCU7 = 7*x^3 + 63*x^2 + 273*x + 441
SOCU8 = 8*x^3 + 84*x^2 + 420*x + 784
SOCU9 = 9*x^3 + 108*x^2 + 612*x + 1296

The 'A' parameters are given by A001477 a(n) = n+1 ; The nonnegative integers.
The 'B' parameters are given by A045943 a(n) = 3*n*(n+1)/2 ; Triangular matchstick numbers.
The 'C' parameters are given by A059270 a(n) = n*(n+1)*(2*n+1)/2 ; a(n) is both the sum of n+1 consecutive integers and the sum of the n immediately higher consecutive integers.
The 'D' parameters are given by A000537 a(n) = (n*(n+1)/2)^2 ; Sum of first n cubes; or n-th triangular number squared.

In case of e.g. SOCU9 with 9 terms (m=9) substitute n in the above formulas with m–1 and we get the following formulae according to the number of terms:
'A' = m
'B' = (3*m^2–3*m)/2
'C' = (2*m^3–3*m^2+m)/2
'D' = ((m^2–m)/2)^2

Sums of TWO cubed consecutives of the form x^3 + (x+1)^3 can only start or end with a 1, 5 or 9.
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
5 can be followed by any number : 50, 51, 52, 53, 54, 55, 56, 57, 58 or 59
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99
There are so far only palindromic sums of cubes of type [SOCU2] of lengths 1 and 4.

Sums of THREE cubed consecutives of the form 3*x^3 + 9*x^2 + 15*x + 9 can only start or end with a 0, 1, 4, 5, 6 or 9.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
4 can only be followed by an even number : 40, 42, 44, 46 or 48
5 can be followed by any number : 50, 51, 52, 53, 54, 55, 56, 57, 58 or 59
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99
There are so far only palindromic sums of cubes of type [SOCU3] of lengths 1, 2 and 5.

Sums of FOUR cubed consecutives of the form 4*x^3 + 18*x^2 + 42*x + 36 can only start or end with a 0, 2, 4, 6 or 8.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
2 can only be followed by an odd number : 21, 23, 25, 27 or 29
4 can only be followed by an even number : 40, 42, 44, 46 or 48
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
8 can only be followed by an even number : 80, 82, 84, 86 or 88
There is so far only one palindromic sum of cubes of type [SOCU4] of length 6.

Sums of FIVE cubed consecutives of the form 5*x^3 + 30*x^2 + 90*x + 100 can only start or end with a 0 or 5.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
5 can be followed by any number : 50, 51, 52, 53, 54, 55, 56, 57, 58 or 59
There is so far no palindromic sum of cubes found of type [SOCU5].

Sums of SIX cubed consecutives of the form 6*x^3 + 45*x^2 + 165*x + 225 can only start or end with a 1, 3, 5, 7 or 9.
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
3 can be followed by any number : 30, 31, 32, 33, 34, 35, 36, 37, 38 or 39
5 can only be followed by 2 or 7 : 52 or 57
7 can be followed by any number : 70, 71, 72, 73, 74, 75, 76, 77, 78 or 79
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99
There is so far no palindromic sum of cubes found of type [SOCU6].

Sums of SEVEN cubed consecutives of the form 7*x^3 + 63*x^2 + 273*x + 441 can only start or end with a 0, 1, 4, 5, 6 or 9.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
4 can only be followed by an even number : 40, 42, 44, 46 or 48
5 can be followed by any number : 50, 51, 52, 53, 54, 55, 56, 57, 58 or 59
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99
There is so far no palindromic sum of cubes found of type [SOCU7].

Sums of EIGHT cubed consecutives of the form 8*x^3 + 84*x^2 + 420*x + 784 can only start or end with a 0, 4 or 6.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
4 can only be followed by an even number : 40, 42, 44, 46 or 48
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
There is so far no palindromic sum of cubes found of type [SOCU8].

Sums of NINE cubed consecutives of the form 9*x^3 + 108*x^2 + 612*x + 1296 can start or end with any digit 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9.
Alas, my palindromes may not have leading 0's! So the zero option must not be investigated.
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
2 can only be followed by an odd number : 21, 23, 25, 27 or 29
3 can be followed by any number : 30, 31, 32, 33, 34, 35, 36, 37, 38 or 39
4 can only be followed by an even number : 40, 42, 44, 46 or 48
5 can only be followed by 2 or 7 : 52 or 57
6 can only be followed by an odd number : 61, 63, 65, 67 or 69
7 can be followed by any number : 70, 71, 72, 73, 74, 75, 76, 77, 78 or 79
8 can only be followed by an even number : 80, 82, 84, 86 or 88
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99
There is so far no palindromic sum of cubes found of type [SOCU9].

A nice coincidence occurred with TWO Cubed Consecutive Integers

16 + 17 = 33
162 + 172 = 545
163 + 173 = 9009

A nice coincidence occurred with FOUR Cubed Consecutive Integers

59 + 60 + 61 + 62 = 242
593 + 603 + 613 + 623 = 886688

Sources Revealed

Huen Y.K. from Singapore developed a general generating function for palindromic sums and products
of consecutive integers using concise programcode written for Macsyma 2.2.1.
Global Generating Function For Palindromic Sums and Products of Consecutive Integers.

 Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online : Neil Sloane's Integer Sequences The regular numbers of form n3 + (n+1)3 [sums of two cubed consecutives] : %N Centered cube numbers: n^3 + (n+1)^3. under A005898. The regular numbers of form n3 + (n+1)3 + (n+2)3 [sums of three cubed consecutives] : %N n^3 + (n+1)^3 + (n+2)^3. under A027602. The regular numbers of form n3 + (n+1)3 + (n+2)3 + (n+3)3 [sums of four cubed consecutives] : %N n^3 + (n+1)^3 + (n+2)^3 + (n+3)^3. under A027603. The regular numbers of form n3 + (n+1)3 + (n+2)3 + (n+3)3 + (n+4)3 [sums of five cubed consecutives] : %N n^3 + (n+1)^3 + (n+2)^3 + (n+3)^3 + (n+4)^3. under A027604. Click here to view some of the author's [P. De Geest] entries to the table. Click here to view some entries to the table about palindromes.

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The Table

Index NrBase Square ExpressionLength
Palindromic Sums of Cubes of Consecutive IntegersLength

[SOCU2] Sums of Cubes of TWO Consecutive Integers
Searched for palindromes up to length 29.
3163 + 1732
9.0094
213 + 231
91
103 + 131
11

[SOCU3] Sums of Cubes of THREE Consecutive Integers
Searched for palindromes up to length 29.
3163 + 173 + 1832
14.8415
223 + 33 + 431
992
103 + 13 + 231
91

[SOCU4] Sums of Cubes of FOUR Consecutive Integers
Searched for palindromes up to length 29.
1593 + 603 + 613 + 6232
886.6886

[SOCU5] Sums of Cubes of FIVE Consecutive Integers
Searched for palindromes up to length 29.
1?3 + ?3 + ?3 + ?3 + ?3?
??

[SOCU6] Sums of Cubes of SIX Consecutive Integers
Searched for palindromes up to length 29.
1?3 + ?3 + ?3 + ?3 + ?3 + ?3?
??

[SOCU7] Sums of Cubes of SEVEN Consecutive Integers
Searched for palindromes up to length 19.
1?3 + ?3 + ?3 + ?3 + ?3 + ?3 + ?3?
??

[SOCU8] Sums of Cubes of EIGHT Consecutive Integers
Searched for palindromes up to length 19.
1?3 + ?3 + ?3 + ?3 + ?3 + ?3 + ?3 + ?3?
??

[SOCU9] Sums of Cubes of NINE Consecutive Integers
Searched for palindromes up to length 19.
11.7223 + 1.7233 + 1.7243 + 1.7253 + 1.7263 + 1.7273 + 1.7283 + 1.7293 + 1.73034
46.277.277.26411

Deep Searching

Searching palindromes equal to sums of cubes of consecutive integers seems very difficult.
Here is an overview of all the palindromes I could find so far with terms m from 2 up to 500000 and
with starting values of the consecutives x from 0 up to 3000000 for the first values m up to 50000.
With this limit we can arrive at around 25-digit palindromes (~ PL = digitlength(3000000^3 * m) ~).
From index 49 (m > 50000) on the startvalue x was searched only from 0 and 500000.

IndexParameters A, B, C, D of the cubic equationSUCO(#terms=A)StartvaluePalindromeLengthRecord & Date
1A=2 B=3 C=3 D=1SUCO2011
2A=2 B=3 C=3 D=1SUCO2191
3A=2 B=3 C=3 D=1SUCO21690094
4A=3 B=9 C=15 D=9SUCO3091
5A=3 B=9 C=15 D=9SUCO32992
6A=3 B=9 C=15 D=9SUCO316148415
7A=4 B=18 C=42 D=36SUCO4598866886
8A=9 B=108 C=612 D=1296SUCO917224627727726411
9A=11 B=165 C=1155 D=3025SUCO11161088016
10A=11 B=165 C=1155 D=3025SUCO11378288286
11A=11 B=165 C=1155 D=3025SUCO112261356665319
12A=16 B=360 C=3720 D=14400SUCO164321121127
13A=19 B=513 C=6327 D=29241SUCO19132399326
14A=21 B=630 C=8610 D=44100SUCO213761675
15A=23 B=759 C=11385 D=64009SUCO234746636647
16A=24 B=828 C=12972 D=76176SUCO242906588088569
17A=46 B=3105 C=94185 D=1071225SUCO462871525177
18A=62 B=5673 C=232593 D=3575881SUCO6230173333718
19A=77 B=8778 C=447678 D=8561476SUCO7715175115718
20A=83 B=10209 C=561495 D=11580409SUCO8343612002168
21A=88 B=11484 C=669900 D=14653584SUCO8827428448248
22A=91 B=12285 C=741195 D=16769025SUCO913073275906660957213
23A=261 B=101790 C=17677530 D=1151244900SUCO2613282711050117211
24A=333 B=165834 C=36759870 D=3055657284SUCO33376695344359610
25A=393 B=231084 C=60466980 D=5933312784SUCO39316696155169610
26A=402 B=241803 C=64722603 D=6496521201SUCO4021483194541614549113
27A=481 B=346320 C=110937840 D=13326393600SUCO48168440516226150412
28A=630 B=594405 C=249451965 D=39257478225SUCO63062958811441188512
29A=909 B=1238058 C=749850462 D=170309734596SUCO909921482325152515232815
30A=1111 B=1849815 C=1369479705 D=380201726025SUCO11116948386776838412
31A=1195 B=2140245 C=1704348435 D=508960962225SUCO1195852273333722512
32A=1750 B=4591125 C=5354782125 D=2342047640625SUCO1750388521308880312513
33A=2063 B=6380859 C=8773681125 D=4523929064209SUCO20634020814477555225557744118
34A=3108 B=14484834 C=30007747770 D=23312268445284SUCO3108336740719273729170415
35A=4329 B=28103868 C=81098395092 D=87758599617936SUCO4329104341523211056465011232519
36A=5656 B=47977020 C=180889357740 D=255754938675600SUCO5656137661023272723201615
37A=6890 B=71197815 C=327011564295 D=563236540086025SUCO68905372544273422437244516
38A=7124 B=76116378 C=361476679122 D=643744777759876SUCO7124104932104780252087401217
39A=15477 B=359283078 C=3706963037778 D=14342703348572676SUCO1547711631916420242024619117
40A=24934 B=932519133 C=15500643868437 D=96621325934563521SUCO24934248514128156446518214118
41A=25297 B=959869368 C=16187556978408 D=102372133736079936SUCO25297171513308458888548033118
42A=53025 B=4217396400 C=149083556941200 D=1976270266081440000SUCO5302516147570633758285733607519
43A=54367 B=4433574483 C=160691951420013 D=2184064744034301921SUCO5436717938148747917356537197478421
44A=62763 B=5908697109 C=247229734869075 D=3879189058433884209SUCO627634534512744458185444721519
45A=66146 B=6562840755 C=289401588773235 D=4785653197276552225SUCO6614611784921553352325335512919
46A=78172 B=9166175118 C=477689105491158 D=9335418477093590436SUCO78172356231487653122111122135678422
47A=85187 B=10885109673 C=618176263439343 D=13165068065915351881SUCO8518712967946215341820281435126421
48A=122174 B=22389546153 C=1823606144615697 D=55699086326368566601SUCO1221741977100103538172749894727183530125Record Largest [ September 2, 2023 ]
49A=131455 B=25920428355 C=2271571299461565 D=74652067345187556225SUCO131455281638570706132733723160707522
50A=139204 B=29066421618 C=2697431747800842 D=93872985075037526436SUCO1392046174240398494702074948930421
51A=203085 B=61864971210 C=8375877830464830 D=425252740312680984900SUCO2030851138552893063041403603982521
52A=237454 B=84576246993 C=13388617243568217 D=794793506157882393561SUCO2374542969851845053933656339350548123
53A=257848 B=99727999884 C=17143076966726460 D=1105074884540344890384SUCO2578484827436162906349989943609261623
54A=351192 B=185003204508 C=43314368597314188 D=3802909519803207946896SUCO3511923224314877418980626089814778423

You might be interested in my Pari/gp program which will recreate the above list given enough running time (days, even weeks).

```{
for(m=2,50000, A=m; B=(3*m^2-3*m)/2; C=(2*m^3-3*m^2+m)/2; D=((m^2-m)/2)^2;
for(x=0,3000000, n=digits(A*x^3+B*x^2+C*x+D);
if(n==Vecrev(n), write("C:/Pari_gp/suco_results.txt", "m=",m," x=",x," n=",n);
print; print1("A=",A," B=",B," C=",C," D=",D);
print("  SUCO",A," ",x," ",fromdigits(n)))); print1(Strchr(13),m));
}
```

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