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Palindromic Quasi_Under_Squares
of the form n+(n+1)^2
rood Sums of Squares rood Sums of Cubes rood Sums of Primes rood comments



Introduction

Palindromic numbers are numbers which read the same from
 p_right left to right (forwards) as from the right to left (backwards) p_left
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


flash

So far I compiled 162 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.



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


Palindromes Quasi_Under_Squares can only end with one of the following digits 1, 5 or 9 !

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 ].





The Table


Index Nr Base Square Expression 
Palindromic Sums of Powers of Consecutive Integers Length 
   
Sums of Powers [1,2] of TWO Consecutive Integers
or Quasi_Under_Squares
162 309.199.492.127.107.420.302.93824
PRIME   95.604.325.931.661.163.590.355.055.309.536.116.613.952.340.65947
161 234.961.102.038.378.693.528.44624
55.206.719.471.089.404.251.001.510.015.240.498.017.491.760.25547
160 100.000.000.000.000.000.000.00024
10.000.000.000.000.000.000.000.300.000.000.000.000.000.000.00147
159 43.127.285.032.036.605.394.66023
1.859.962.714.234.528.623.602.552.063.268.254.324.172.699.58146
158 10.000.000.000.000.000.000.00023
100.000.000.000.000.000.000.030.000.000.000.000.000.000.00145
157 7.072.577.474.842.463.777.27622
50.021.352.137.649.001.344.377.344.310.094.673.125.312.00544
156 3.171.834.144.204.063.497.99722
10.060.531.838.338.723.877.033.077.832.783.383.813.506.00144
155 3.152.274.117.846.295.426.39922
9.936.832.114.043.640.025.671.765.200.463.404.112.386.39943
154 2.361.847.617.370.644.335.25122
5.578.324.167.679.389.569.078.709.659.839.767.614.238.75543
153 2.249.051.604.425.933.192.29622
5.058.233.119.370.864.274.848.484.724.680.739.113.328.50543
152 PRIME   2.245.300.889.691.072.846.76122
5.041.376.085.247.523.275.876.785.723.257.425.806.731.40543
151 PRIME   1.334.722.262.459.512.275.40722
1.781.483.517.905.039.171.511.151.719.305.097.153.841.87143
150 1.016.914.646.505.040.831.89722
1.034.115.398.276.472.153.826.283.512.746.728.935.114.30143
149 1.293.195.253.540.153.504.24522
1.672.353.963.778.781.904.457.544.091.878.773.693.532.76143
148 1.000.000.000.000.000.000.00022
1.000.000.000.000.000.000.003.000.000.000.000.000.000.00143
147 988.096.222.636.405.555.27921
976.334.145.188.333.134.069.960.431.333.881.541.433.67942
146 411.230.995.850.242.819.47721
169.110.931.947.982.427.014.410.724.289.749.139.011.96142
145 PRIME   343.920.240.954.164.126.02721
118.281.132.137.970.311.356.653.113.079.731.231.182.81142
144 338.684.647.404.406.346.17021
114.707.290.387.447.050.358.853.050.744.783.092.707.41142
143 120.322.018.279.570.148.31721
14.477.388.082.869.212.914.641.921.296.828.088.377.44141
142 102.874.287.511.911.917.37221
10.583.119.031.083.516.275.257.261.538.013.091.138.50141
141 100.000.000.000.000.000.00021
10.000.000.000.000.000.000.300.000.000.000.000.000.00141
140 98.083.666.304.811.516.62820
9.620.405.595.793.618.072.992.708.163.975.955.040.26940
139 97.180.618.679.137.573.82420
9.444.072.646.859.942.723.993.272.499.586.462.704.44940
138 30.599.051.923.144.257.95820
936.301.978.595.278.311.515.113.872.595.879.103.63939
137 30.580.227.456.633.992.31420
935.150.311.299.471.490.363.094.174.992.113.051.53939
136 22.436.877.620.839.787.45620
503.413.477.372.541.281.222.182.145.273.774.314.30539
135 22.370.394.311.827.201.19620
500.434.541.666.630.798.646.897.036.666.145.434.00539
134 13.960.454.038.250.698.15520
194.894.276.954.110.225.626.522.011.459.672.498.49139
133 12.461.759.392.637.263.97520
155.295.447.159.983.070.353.070.389.951.744.592.55139
132 12.571.979.434.964.434.92220
158.054.666.913.168.672.404.276.861.319.666.450.85139
131 11.880.465.900.990.034.70520
141.145.470.024.586.957.141.759.685.420.074.541.14139
130 11.495.287.842.685.363.66220
132.141.642.586.189.922.141.229.981.685.246.141.23139
129 10.000.000.000.000.000.00020
100.000.000.000.000.000.030.000.000.000.000.000.00139
128 3.166.927.238.501.507.87219
10.029.428.133.962.786.533.568.726.933.182.492.00138
127 3.129.349.516.433.813.41819
9.792.828.396.004.541.884.881.454.006.938.282.97937
126 1.000.000.000.000.000.00019
1.000.000.000.000.000.003.000.000.000.000.000.00137
125 984.253.755.175.473.84418
968.755.454.577.021.607.706.120.775.454.557.86936
124 314.323.549.693.580.56418
98.799.293.891.972.811.211.827.919.839.299.78935
123 100.089.245.408.957.00018
10.017.857.046.534.420.202.443.564.075.871.00135
122 100.000.000.000.000.00018
10.000.000.000.000.000.300.000.000.000.000.00135
121 96.478.719.637.704.43917
9.308.143.342.930.776.446.770.392.433.418.03934
120 39.213.318.281.545.04717
1.537.684.330.649.755.115.579.460.334.867.35134
119 39.077.257.615.763.87517
1.527.032.062.768.775.885.778.672.602.307.25134
118 35.809.584.980.401.74017
1.282.326.376.468.613.993.168.646.736.232.82134
117 31.248.325.265.075.65417
976.457.831.871.965.535.569.178.138.754.67933
116 10.000.000.000.000.00017
PRIME   100.000.000.000.000.030.000.000.000.000.00133
115 3.849.104.088.659.51716
14.815.602.285.335.422.453.358.220.651.84132
114 3.752.868.400.303.99216
14.084.021.230.000.255.200.003.212.048.04132
113 3.473.560.140.754.73216
12.065.620.051.440.044.004.415.002.656.02132
112 2.358.576.798.224.28616
5.562.884.513.121.931.391.213.154.882.65531
111 1.412.890.844.107.34216
1.996.260.537.362.361.632.637.350.626.99131
110 1.400.359.109.515.44216
PRIME   1.961.005.635.602.885.882.065.365.001.69131
109 1.397.804.508.310.23016
1.953.857.443.452.408.042.543.447.583.59131
108 1.220.422.583.985.39216
1.489.431.283.501.584.851.053.821.349.84131
107 1.142.090.725.619.31016
1.304.371.225.545.645.465.455.221.734.03131
106 1.049.921.608.476.00216
PRIME   1.102.335.383.944.838.384.493.835.332.01131
105 1.003.691.398.020.99716
1.007.396.422.461.346.431.642.246.937.00131
104 1.000.000.000.000.00016
1.000.000.000.000.003.000.000.000.000.00131
103 430.721.068.556.66015
185.520.638.898.592.295.898.836.025.58130
102 309.637.329.125.36315
95.875.275.587.889.298.878.557.257.85929
101 123.245.184.901.44715
15.189.375.601.392.229.310.657.398.15129
100 122.258.874.462.69215
14.947.232.384.884.648.848.323.274.94129
99 103.594.624.078.97215
10.731.846.138.063.836.083.164.813.70129
98 100.000.000.000.00015
PRIME   10.000.000.000.000.300.000.000.000.00129
97 71.003.776.901.96114
5.041.536.334.343.663.434.336.351.40528
96 31.375.435.997.60814
PRIME   984.417.984.040.090.040.489.714.48927
95 10.000.000.000.00014
100.000.000.000.030.000.000.000.00127
94 4.342.164.720.88513
18.854.394.463.311.336.449.345.88126
93 3.031.211.772.41913
9.188.244.809.260.629.084.428.81925
92 2.346.319.275.00613
5.505.214.140.271.720.414.125.05525
91 1.389.789.688.54213
1.931.515.378.381.838.735.151.39125
90 1.361.229.261.21213
1.852.945.101.583.851.015.492.58125
89 1.233.391.513.74713
1.521.254.626.186.816.264.521.25125
88 1.132.222.222.13213
1.281.927.160.292.920.617.291.82125
87 1.000.000.000.00013
1.000.000.000.003.000.000.000.00125
86 713.321.230.88112
508.827.178.427.724.871.728.80524
85 707.787.945.80112
500.963.776.223.322.677.369.00524
84 391.072.107.74712
152.937.393.458.854.393.739.25124
83 330.896.531.69712
109.492.514.690.096.415.294.90124
82 308.614.186.96312
PRIME   95.242.716.395.759.361.724.25923
81 304.837.396.09312
92.925.838.057.675.083.852.92923
80 236.020.325.65112
55.705.594.121.112.149.550.75523
79 234.608.196.39112
55.041.005.814.541.850.014.05523
78 225.562.844.35612
50.878.596.754.645.769.587.80523
77 100.000.000.00012
PRIME   10.000.000.000.300.000.000.00123
76 36.648.687.11011
1.343.126.266.996.626.213.43122
75 13.772.991.53811
PRIME   189.695.295.919.592.596.98121
74 10.000.000.00011
100.000.000.030.000.000.00121
73 3.158.325.17410
9.975.017.914.197.105.79919
72 3.018.331.09410
9.110.322.602.062.230.11919
71 3.004.166.88810
9.025.018.699.968.105.20919
70 2.361.405.22610
5.576.234.648.464.326.75519
69 2.255.496.51610
5.087.264.540.454.627.80519
68 2.253.899.68110
5.080.063.778.773.600.80519
67 2.248.287.09610
5.054.794.872.784.974.50519
66 1.398.162.76710
1.954.859.127.219.584.59119
65 1.229.782.82210
1.512.365.792.975.632.15119
64 1.194.696.28010
1.427.299.205.029.927.24119
63 1.097.411.26510
1.204.311.487.841.134.02119
62 1.059.993.22710
1.123.585.644.465.853.21119
61 1.057.255.30210
1.117.788.776.778.877.11119
60 1.000.000.0001 + 1.000.000.001210
1.000.000.003.000.000.00119
59 340.313.8701 + 340.313.8712 
115.813.531.135.318.51118
58 326.732.6721 + 326.732.6732 
106.754.239.932.457.60118
57 307.729.0241 + 307.729.0252 
94.697.153.135.179.64917
56 303.357.9931 + 303.357.9942 
92.026.072.827.062.02917
55 134.606.1371 + 134.606.1382 
18.118.812.521.881.18117
54 106.040.6021 + 106.040.6032 
11.244.609.590.644.21117
53 100.000.0001 + 100.000.0012 
10.000.000.300.000.00117
52 23.615.2261 + 23.615.2272 
557.678.969.876.75515
51 11.373.4401 + 11.373.4412 
129.355.171.553.92115
50 10.946.6701 + 10.946.6712 
PRIME   119.829.616.928.91115
49 10.000.0001 + 10.000.0012 
100.000.030.000.00115
48 3.267.3251 + 3.267.3262 
10.675.422.457.60114
47 3.126.7541 + 3.126.7552 
9.776.599.956.77913
46 3.040.4041 + 3.040.4052 
9.244.065.604.42913
45 2.356.5311 + 2.356.5322 
5.553.245.423.55513
44 2.346.3661 + 2.346.3672 
5.505.440.445.05513
43 1.334.1071 + 1.334.1082 
1.779.845.489.77113
42 1.202.8051 + 1.202.8062 
PRIME   1.446.743.476.44113
41 1.172.8351 + 1.172.8362 
1.375.545.455.73113
40 1.099.2651 + 1.099.2662 
1.208.386.838.02113
39 1.060.6021 + 1.060.6032 
1.124.879.784.21113
38 1.000.0001 + 1.000.0012 
1.000.003.000.00113
37 987.4431 + 987.4442 
975.046.640.57912
36 709.7811 + 709.7822 
503.791.197.30512
35 308.8631 + 308.8642 
95.397.279.35911
34 236.0261 + 236.0272 
55.708.980.75511
33 234.5661 + 234.5672 
55.021.912.05511
32 110.6651 + 110.6662 
12.247.074.22111
31 100.0001 + 100.0012 
10.000.300.00111
30 96.4391 + 96.4402 
9.300.770.03910
29 95.6031 + 95.6042 
9.140.220.41910
28 32.6721 + 32.6732 
1.067.557.60110
27 31.4741 + 31.4752 
PRIME   990.707.0999
26 30.8341 + 30.8352 
950.828.0599
25 30.4041 + 30.4052 
924.494.4299
24 13.2071 + 13.2082 
174.464.4719
23 12.6521 + 12.6532 
160.111.0619
22 10.6021 + 10.6032 
PRIME   112.434.2119
21 10.3251 + 10.3262 
106.636.6019
20 10.0001 + 10.0012 
PRIME   100.030.0019
19 3.0731 + 3.0742 
9.452.5497
18 1.3071 + 1.3082 
PRIME   1.712.1717
17 1.1321 + 1.1332 
1.284.8217
16 1.0001 + 1.0012 
PRIME   1.003.0017
15 4421 + 4432 
196.6916
14 3801 + 3812 
145.5416
13 3251 + 3262 
106.6016
12 1221 + 1232 
15.2515
11 1171 + 1182 
14.0415
10 1001 + 1012 
PRIME   10.3015
9 941 + 952 
9.1194
8 351 + 362 
1.3314
7 291 + 302 
PRIME   9293
6 211 + 222 
5053
5 121 + 132 
PRIME   1813
4 101 + 112 
PRIME   1313
3 61 + 72 
552
2 21 + 32 
PRIME   112
1 11 + 22 
PRIME   51
0 01 + 12 
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
6 100.010.0002 + 100.010.00139
1.000.300.070.009.000.700.030.00125
5 1.001.0002 + 1.001.00137
1.003.007.009.007.003.00119
4 10.1002 + 10.10135
PRIME   1.030.709.070.30113
3 1102 + 11133
1.379.7317
2 42 + 531
1413
1 12 + 231
91
0 02 + 131
11
   
Sums of Powers [3,4] of TWO Consecutive Integers
Searched upto basenumber 1.000.000.000
1 ?3 + ?4 
??
0 03 + 141
11
   
Sums of Powers [4,5] of TWO Consecutive Integers
Searched upto basenumber 1.000.000.000
1 14 + 251
332
0 04 + 151
11
   
Sums of Powers of THREE Consecutive Integers
1 ?1 + ?2 + ?3 
??
0 01 + 12 + 231
91
   
Sums of Powers of FOUR Consecutive Integers
1 ?1 + ?2 + ?3 + ?4 
??
   
Sums of Powers of FIVE Consecutive Integers
1 ?1 + ?2 + ?3 + ?4 + ?5 
??


In case you are also interested in the primes of these numberforms
check out the following entries in Sloane's table :

%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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E-mail address : pdg@worldofnumbers.com