HOME plateWON | World!OfNumbers Palindromic Quasi_Under_Squaresof the form n+(n+1)^2 Sums of Squares Sums of Cubes Sums of Primes 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 Powers of Consecutive Integers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

 basen + ( base + 1 )n+1 + ...

Let me name the numbers of the form n + (n+1)^2 or (n-1) + n^2 as follows : Quasi_Under_Squares.
Numbers of the form n^2 + (n+1) then become the Quasi_Over_Squares.
Readers with more original suggestions for naming these numbers can always send their proposals to me at my e-mail address.

Palindromic Quasi_Under_Squares of the form n+(n+1)^2

Index 33 shows a particularly beautiful number in its most significant part.
2345661 + 2345672 = 55021912055
The digits from 2 to 7 are arranged in ascending order !

Index 45 shows a palindromic basenumber in its most significant part.
23565311 + 23565322 = 5553245423555
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So far I compiled 163 Palindromic Quasi_Under_Squares.
Here is the largest one that Feng Yuan discovered during [ January 15-24, 2008 ]

 This basenumber 309.199.492.127.107.420.302.938 has 24 digits yielding the following Prime Palindromic Quasi_Under_Square 95.604.325.931.661.163.590.355.055.309.536.116.613.952.340.659 with a length of 47 digits.

Here is the largest one that I discovered on [ April 3, 2005 ]

 This basenumber 39.213.318.281.545.047 has 17 digits yielding the following Palindromic Quasi_Under_Square 1.537.684.330.649.755.115.579.460.334.867.351 with a length of 34 digits.

Palindromic Quasi_Under_Squares can only end with one of the following digits 1, 5 or 9 !
1 can be followed by any number : 10, 11, 12, 13, 14, 15, 16, 17, 18 or 19
5 can only be followed by 0 or 5 : 50 or 55
9 can be followed by any number : 90, 91, 92, 93, 94, 95, 96, 97, 98 or 99

Sources Revealed

 Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online : Neil Sloane's Integer Sequences The normal numbers of the form n + (n+1)^2 are already categorised. %N Numbers of the form n + (n+1)^2 under A028387 . Check out the following two entries about Palindromic Powers %N n + (n+1)^2 is a palindrome under A028348. %N Palindromes of form n + (n+1)^2 under A028349. 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.

If one searches the database for the short sequence {5,11,19,29,41} you get a positive reply and the entry shown is :
%N Primes of form n^2 – n – 1 under A002327.
( %N n^2 – n – 1 is prime. under A002328. )
Careful observation shows that this is just a subset of the numbers of the form n + (n+1)^2 !

I will give a proof that every number of the form n^2 – n – 1 is also a member of the numbers of the form n + (n+1)^2
whereby its basenumbers are two units displaced.
Substitute n with m+2 and we derive :

n^2 – n – 1 [or n^2 – (n+1)] =
(m+2)^2 – (m+2) – 1 =
(m^2 + 4m + 4) – m – 3 =
m^2 + 3m + 1 =
m + (m^2 + 2m + 1) =
m + (m+1)^2
Another formula that is related to the above one is the Quasi_Over_Square n^2 + (n+1).
Substitute n with m+1 and subtract 2 from the whole to arrive at the original sequence !
The constant value 2 comes from the fact that we can rewrite n + (n+1)^2 as n^2 + (n–1) or n^2 + n – 1.
And n^2 + n – 1 differs exactly 2 from n^2 + n + 1 !
n^2 + (n+1) =
(m+1)^2 + ((m+1)+1) – 2 =
(m^2+2m+1)+(m+2) – 2 =
m^2+3m+1 =
m + (m^2+2m+1) =
m + (m+1)^2
Due to the subtraction of the constant 2 we end up with a different integer sequence.
The palindromic numbers resulting from the formula n^2 + (n+1) are displayed at the following page :
Quasi-Over-Squares of the form n^2 + (n+1)

Not long after I submitted the numbers of the form n + (n+1)^2 under A028387, Neil Sloane pointed out to me that his Integer Sequence A028392 is the complement of the former. This means that what's missing in the first sequence appears in the second. The formula of the complement series is n + [sqrt(n)]. The square brackets indicate that the integer part of the real number in it must be considered.

0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,
31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58

When one subtracts the second part [sqrt(n)] from n ( See John Mellor's entry A028391 ) then another curiosity shows up that also relates n – [sqrt(n)] to the numbers of the form n + (n+1)^2. Each number preceding the couple of two identical numbers is also of the form n + (n+1)^2. See for yourself :
0,0,1,2,2,3,4,5,6,6,7,8,9,10,11,12,12,13,14,15,16,17,18,19,20,20,21,22,23,24,25,
26,27,28,29,30,30,31,32,33,34,35,36,37,38,39,40,41,42,42,43,44,45,46,47,48,49,50

One infinite expanding patterns immediately shows up :
10 + 112 = 131
100 + 1012 = 10301
1000 + 10012 = 1003001
10000 + 100012 = 100030001

There exist no such palindromes of length 8, 16, 20.

My search program searched for palindromes exhaustively upto  length 33

From length 34 on these numbers were discovered by Feng Yuan during [ January 15-24, 2008 ].

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The Table

Index NrBase Square ExpressionLength
Palindromic Sums of Powers of Consecutive IntegersLength

Sums of Powers [1,2] of TWO Consecutive Integers
or Quasi_Under_Squares
163309.199.492.127.107.420.302.93824
Prime!    95.604.325.931.661.163.590.355.055.309.536.116.613.952.340.65947
162234.961.102.038.378.693.528.44624
55.206.719.471.089.404.251.001.510.015.240.498.017.491.760.25547
161100.000.000.000.000.000.000.00024
10.000.000.000.000.000.000.000.300.000.000.000.000.000.000.00147
16043.127.285.032.036.605.394.66023
1.859.962.714.234.528.623.602.552.063.268.254.324.172.699.58146
15910.000.000.000.000.000.000.00023
100.000.000.000.000.000.000.030.000.000.000.000.000.000.00145
1587.072.577.474.842.463.777.27622
50.021.352.137.649.001.344.377.344.310.094.673.125.312.00544
1573.171.834.144.204.063.497.99722
10.060.531.838.338.723.877.033.077.832.783.383.813.506.00144
1563.152.274.117.846.295.426.39922
9.936.832.114.043.640.025.671.765.200.463.404.112.386.39943
1552.361.847.617.370.644.335.25122
5.578.324.167.679.389.569.078.709.659.839.767.614.238.75543
1542.249.051.604.425.933.192.29622
5.058.233.119.370.864.274.848.484.724.680.739.113.328.50543
153Prime!    2.245.300.889.691.072.846.76122
5.041.376.085.247.523.275.876.785.723.257.425.806.731.40543
152Prime!    1.334.722.262.459.512.275.40722
1.781.483.517.905.039.171.511.151.719.305.097.153.841.87143
1511.016.914.646.505.040.831.89722
1.034.115.398.276.472.153.826.283.512.746.728.935.114.30143
1501.293.195.253.540.153.504.24522
1.672.353.963.778.781.904.457.544.091.878.773.693.532.76143
1491.000.000.000.000.000.000.00022
1.000.000.000.000.000.000.003.000.000.000.000.000.000.00143
148988.096.222.636.405.555.27921
976.334.145.188.333.134.069.960.431.333.881.541.433.67942
147411.230.995.850.242.819.47721
169.110.931.947.982.427.014.410.724.289.749.139.011.96142
146Prime!    343.920.240.954.164.126.02721
118.281.132.137.970.311.356.653.113.079.731.231.182.81142
145338.684.647.404.406.346.17021
114.707.290.387.447.050.358.853.050.744.783.092.707.41142
144120.322.018.279.570.148.31721
14.477.388.082.869.212.914.641.921.296.828.088.377.44141
143102.874.287.511.911.917.37221
10.583.119.031.083.516.275.257.261.538.013.091.138.50141
142100.000.000.000.000.000.00021
10.000.000.000.000.000.000.300.000.000.000.000.000.00141
14198.083.666.304.811.516.62820
9.620.405.595.793.618.072.992.708.163.975.955.040.26940
14097.180.618.679.137.573.82420
9.444.072.646.859.942.723.993.272.499.586.462.704.44940
13930.599.051.923.144.257.95820
936.301.978.595.278.311.515.113.872.595.879.103.63939
13830.580.227.456.633.992.31420
935.150.311.299.471.490.363.094.174.992.113.051.53939
13722.436.877.620.839.787.45620
503.413.477.372.541.281.222.182.145.273.774.314.30539
13622.370.394.311.827.201.19620
500.434.541.666.630.798.646.897.036.666.145.434.00539
13513.960.454.038.250.698.15520
194.894.276.954.110.225.626.522.011.459.672.498.49139
13412.461.759.392.637.263.97520
155.295.447.159.983.070.353.070.389.951.744.592.55139
13312.571.979.434.964.434.92220
158.054.666.913.168.672.404.276.861.319.666.450.85139
13211.880.465.900.990.034.70520
141.145.470.024.586.957.141.759.685.420.074.541.14139
13111.495.287.842.685.363.66220
132.141.642.586.189.922.141.229.981.685.246.141.23139
13010.000.000.000.000.000.00020
100.000.000.000.000.000.030.000.000.000.000.000.00139
1293.166.927.238.501.507.87219
10.029.428.133.962.786.533.568.726.933.182.492.00138
1283.129.349.516.433.813.41819
9.792.828.396.004.541.884.881.454.006.938.282.97937
1271.000.000.000.000.000.00019
1.000.000.000.000.000.003.000.000.000.000.000.00137
126984.253.755.175.473.84418
968.755.454.577.021.607.706.120.775.454.557.86936
125314.323.549.693.580.56418
98.799.293.891.972.811.211.827.919.839.299.78935
124100.089.245.408.957.00018
10.017.857.046.534.420.202.443.564.075.871.00135
123100.000.000.000.000.00018
10.000.000.000.000.000.300.000.000.000.000.00135
12296.478.719.637.704.43917
9.308.143.342.930.776.446.770.392.433.418.03934
12139.213.318.281.545.04717
1.537.684.330.649.755.115.579.460.334.867.35134
12039.077.257.615.763.87517
1.527.032.062.768.775.885.778.672.602.307.25134
11935.809.584.980.401.74017
1.282.326.376.468.613.993.168.646.736.232.82134
11831.248.325.265.075.65417
976.457.831.871.965.535.569.178.138.754.67933
11710.000.000.000.000.00017
Prime!    100.000.000.000.000.030.000.000.000.000.00133
1163.849.104.088.659.51716
14.815.602.285.335.422.453.358.220.651.84132
1153.752.868.400.303.99216
14.084.021.230.000.255.200.003.212.048.04132
1143.473.560.140.754.73216
12.065.620.051.440.044.004.415.002.656.02132
1132.358.576.798.224.28616
5.562.884.513.121.931.391.213.154.882.65531
1121.412.890.844.107.34216
1.996.260.537.362.361.632.637.350.626.99131
1111.400.359.109.515.44216
Prime!    1.961.005.635.602.885.882.065.365.001.69131
1101.397.804.508.310.23016
1.953.857.443.452.408.042.543.447.583.59131
1091.220.422.583.985.39216
1.489.431.283.501.584.851.053.821.349.84131
1081.142.090.725.619.31016
1.304.371.225.545.645.465.455.221.734.03131
1071.049.921.608.476.00216
Prime!    1.102.335.383.944.838.384.493.835.332.01131
1061.003.691.398.020.99716
1.007.396.422.461.346.431.642.246.937.00131
1051.000.000.000.000.00016
1.000.000.000.000.003.000.000.000.000.00131
104430.721.068.556.66015
185.520.638.898.592.295.898.836.025.58130
103309.637.329.125.36315
95.875.275.587.889.298.878.557.257.85929
102123.245.184.901.44715
15.189.375.601.392.229.310.657.398.15129
101122.258.874.462.69215
14.947.232.384.884.648.848.323.274.94129
100103.594.624.078.97215
10.731.846.138.063.836.083.164.813.70129
99100.000.000.000.00015
Prime!    10.000.000.000.000.300.000.000.000.00129
9871.003.776.901.96114
5.041.536.334.343.663.434.336.351.40528
9731.375.435.997.60814
Prime Curios!    984.417.984.040.090.040.489.714.48927
9610.000.000.000.00014
100.000.000.000.030.000.000.000.00127
954.342.164.720.88513
18.854.394.463.311.336.449.345.88126
943.031.211.772.41913
9.188.244.809.260.629.084.428.81925
932.346.319.275.00613
5.505.214.140.271.720.414.125.05525
921.389.789.688.54213
1.931.515.378.381.838.735.151.39125
911.361.229.261.21213
1.852.945.101.583.851.015.492.58125
901.233.391.513.74713
1.521.254.626.186.816.264.521.25125
891.132.222.222.13213
1.281.927.160.292.920.617.291.82125
881.000.000.000.00013
1.000.000.000.003.000.000.000.00125
87713.321.230.88112
508.827.178.427.724.871.728.80524
86707.787.945.80112
500.963.776.223.322.677.369.00524
85391.072.107.74712
152.937.393.458.854.393.739.25124
84330.896.531.69712
109.492.514.690.096.415.294.90124
83308.614.186.96312
Prime!    95.242.716.395.759.361.724.25923
82304.837.396.09312
92.925.838.057.675.083.852.92923
81236.020.325.65112
55.705.594.121.112.149.550.75523
80234.608.196.39112
55.041.005.814.541.850.014.05523
79225.562.844.35612
50.878.596.754.645.769.587.80523
78100.000.000.00012
Prime!    10.000.000.000.300.000.000.00123
7736.648.687.11011
1.343.126.266.996.626.213.43122
7613.772.991.53811
Prime!    189.695.295.919.592.596.98121
7510.000.000.00011
100.000.000.030.000.000.00121
743.158.325.17410
9.975.017.914.197.105.79919
733.018.331.09410
9.110.322.602.062.230.11919
723.004.166.88810
9.025.018.699.968.105.20919
712.361.405.22610
5.576.234.648.464.326.75519
702.255.496.51610
5.087.264.540.454.627.80519
692.253.899.68110
5.080.063.778.773.600.80519
682.248.287.09610
5.054.794.872.784.974.50519
671.398.162.76710
1.954.859.127.219.584.59119
661.229.782.82210
1.512.365.792.975.632.15119
651.194.696.28010
1.427.299.205.029.927.24119
641.097.411.26510
1.204.311.487.841.134.02119
631.059.993.22710
1.123.585.644.465.853.21119
621.057.255.30210
1.117.788.776.778.877.11119
611.000.000.0001 + 1.000.000.001210
1.000.000.003.000.000.00119
60340.313.8701 + 340.313.87129
115.813.531.135.318.51118
59326.732.6721 + 326.732.67329
106.754.239.932.457.60118
58307.729.0241 + 307.729.02529
94.697.153.135.179.64917
57303.357.9931 + 303.357.99429
92.026.072.827.062.02917
56134.606.1371 + 134.606.13829
18.118.812.521.881.18117
55106.040.6021 + 106.040.60329
11.244.609.590.644.21117
54100.000.0001 + 100.000.00129
10.000.000.300.000.00117
5323.615.2261 + 23.615.22728
557.678.969.876.75515
5211.373.4401 + 11.373.44128
129.355.171.553.92115
5110.946.6701 + 10.946.67128
Prime!    119.829.616.928.91115
5010.000.0001 + 10.000.00128
100.000.030.000.00115
493.267.3251 + 3.267.32627
10.675.422.457.60114
483.126.7541 + 3.126.75527
9.776.599.956.77913
473.040.4041 + 3.040.40527
9.244.065.604.42913
462.356.5311 + 2.356.53227
5.553.245.423.55513
452.346.3661 + 2.346.36727
5.505.440.445.05513
441.334.1071 + 1.334.10827
1.779.845.489.77113
431.202.8051 + 1.202.80627
Prime!    1.446.743.476.44113
421.172.8351 + 1.172.83627
1.375.545.455.73113
411.099.2651 + 1.099.26627
1.208.386.838.02113
401.060.6021 + 1.060.60327
1.124.879.784.21113
391.000.0001 + 1.000.00127
1.000.003.000.00113
38987.4431 + 987.44426
975.046.640.57912
37709.7811 + 709.78226
503.791.197.30512
36308.8631 + 308.86426
95.397.279.35911
35236.0261 + 236.02726
55.708.980.75511
34234.5661 + 234.56726
55.021.912.05511
33110.6651 + 110.66626
12.247.074.22111
32100.0001 + 100.00126
10.000.300.00111
3196.4391 + 96.44025
9.300.770.03910
3095.6031 + 95.60425
9.140.220.41910
2932.6721 + 32.67325
1.067.557.60110
2831.4741 + 31.47525
Prime!    990.707.0999
2730.8341 + 30.83525
950.828.0599
2630.4041 + 30.40525
924.494.4299
2513.2071 + 13.20825
174.464.4719
2412.6521 + 12.65325
160.111.0619
2310.6021 + 10.60325
Prime!    112.434.2119
2210.3251 + 10.32625
106.636.6019
2110.0001 + 10.00125
Prime!    100.030.0019
203.0731 + 3.07424
9.452.5497
191.3071 + 1.30824
Prime!    1.712.1717
181.1321 + 1.13324
1.284.8217
171.0001 + 1.00124
Prime!    1.003.0017
164421 + 44323
196.6916
153801 + 38123
145.5416
143251 + 32623
106.6016
131221 + 12323
15.2515
121171 + 11823
14.0415
111001 + 10123
Prime!    10.3015
10941 + 9522
9.1194
9351 + 3622
1.3314
8291 + 3022
Prime!    9293
7211 + 2222
5053
6121 + 1322
Prime!    1813
5101 + 1122
Prime!    1313
461 + 721
552
321 + 321
Prime!    112
211 + 221
Prime!    51
101 + 121
11

Sums of Powers [2,3] of TWO Consecutive Integers
?10.000.100.0002 + 10.000.100.001311
1.000.030.000.700.009.000.070.000.300.00131
7100.010.0002 + 100.010.00139
1.000.300.070.009.000.700.030.00125
61.001.0002 + 1.001.00137
1.003.007.009.007.003.00119
510.1002 + 10.10135
Prime!    1.030.709.070.30113
41102 + 11133
1.379.7317
342 + 531
1413
212 + 231
91
102 + 131
11

Sums of Powers [3,2] of TWO Consecutive Integers
Searched upto basenumber 100.000.000
4733 + 7421
394.4936
353 + 621
1613
213 + 221
51
103 + 121
11

Sums of Powers [3,4] of TWO Consecutive Integers
Searched upto basenumber 1.000.000.000
103 + 141
11

Sums of Powers [4,5] of TWO Consecutive Integers
Searched upto basenumber 1.000.000.000
214 + 251
332
104 + 151
11

Sums of Powers [5,6] of TWO Consecutive Integers
Searched upto basenumber 10.000.000
105 + 161
11

Sums of Powers [6,7] of TWO Consecutive Integers
Searched upto basenumber 10.000.000
106 + 171
11

Sums of Powers [7,8] of TWO Consecutive Integers
Searched upto basenumber 10.000.000
107 + 181
11

Sums of Powers [8,9] of TWO Consecutive Integers
Searched upto basenumber 10.000.000
108 + 191
11

Sums of Powers [1,2,3] of THREE Consecutive Integers
Searched upto basenumber 1.000.000.000
101 + 12 + 231
91

Sums of Powers [3,2,1] of THREE Consecutive Integers
Searched upto basenumber 100.000.000
113 + 22 + 31
81
103 + 12 + 21
31

Sums of Powers [1,2,3,4] of FOUR Consecutive Integers
Searched upto basenumber 1.000.000.000
1?1 + ?2 + ?3 + ?4
??

Sums of Powers [4,3,2,1] of FOUR Consecutive Integers
Searched upto basenumber 100.000.000
454 + 63 + 72 + 81
8983
344 + 53 + 62 + 71
4243
214 + 23 + 32 + 41
222
104 + 13 + 22 + 31
81

Sums of Powers [1,2,3,4,5] of FIVE Consecutive Integers
Searched upto basenumber 1.000.000.000
1?1 + ?2 + ?3 + ?4 + ?5
??

Sums of Powers [5,4,3,2,1] of FIVE Consecutive Integers
Searched upto basenumber 100.000.000
105 + 14 + 23 + 32 + 41
222

 In case you are also interested in the primes of these numberforms check out the following entries in Sloane's "Integer Sequences" Encyclopedia : %N n + (n+1)^2 is a prime. under A002328. %N Primes of form n + (n+1)^2. under A002327. %N n + (n+1)^2 + (n+2)^3 + (n+3)^4 + (n+4)^5 is a prime. under A027830. %N Primes of form n + (n+1)^2 + (n+2)^3 + (n+3)^4 + (n+4)^5. under A027886. There are no primes of the form n + (n+1)^2 + (n+2)^3. All such numbers are divisible by (n+3) twice ! The polynomial can be written as (n+1)*(n+3)*(n+3). There are no primes of the form n + (n+1)^2 + (n+2)^3 + (n+3)^4. All such numbers are even !

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