HOME plateWON | World!OfNumbers Palindromic Quasi_Over_Squaresof the form n^2+(n+1) n(n+0) n(n+1) n(n+2) n(n+x) n^2+1 n^2+x n^2–x n^2+(n+x)

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 : 7, 3113, 44611644

Quasi_Over_Square numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

 base2 + ( base + 1 )1

 PLAIN TEXT QUASIMOR

Palindromic Quasi_Over_Squares

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 (coded as QO-SQU) .
Readers with more original suggestions for naming these numbers can always send their proposals to me at my e-mail address.

Just like Warut Roonguthai ( See Palindromic Pronic Numbers n(n+1) ) I also recognized
a similar finite growing pattern in these Quasi_Over_Squares basenumbers.
26 (5445)n 62, for n = 2 to 4
26545 (5445)n 437, for n = 0 to 4
|Astonishing!| the largest known Quasi_Over_Square now has a length of 47 digits !

So far this compilation counts 150 Palindromic Quasi_Over_Squares.
Here is the largest sporadic QM1 (of type x^2+(x+1)) that Patrick De Geest,
discovered, using CUDA code by Robert Xiao, on [ June 20, 2023 ].

 This basenumber 198.673.328.269.067.804.729.345.531 has 27 digits yielding the following Palindromic Quasi_Over_Square 39.471.091.365.508.776.518.863.915.551.936.881.567.780.556.319.017.493 with a length of 53 digits.

All palindromic numbers of the form n^2 + (n+1) can only end with 1, 3 or 7
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
7 can only be followed by 0 or 5 : 70 or 75
There are no palindromic quasi_over_squares of type x^2 + (x + 1) of EVEN length.

 Quasi_Over_Square Change of variables CUDApalin parameters Base Correction x^2 + (x+1) n.a. `A B C →  1 1 1 ` base = CUDAbase

It is no coincidence that I didn't find Palindromic Quasi_Over_Squares of even length.

Actually the following conjecture was proven for me by Neo Chee Beng and Warut Roonguthai almost simultaneously :
Every number of the form n^2+(n+1) is NOT divisible by 11.
Proof : by using mod 11
```        n mod 11 :   0   1   2   3   4   5   6   7   8   9   10
(n^2+n+1) mod 11 :   1   3   7   2  10   9  10   2   7   3    1
[ if zero appeared in the second line then it would be divisible by 11 ]
```
Because Palindromic Quasi_Over_Squares of EVEN length are always divisible by 11 we immediately see that they cannot occur here.

And also that :
Every number of the form n^2+(n+1) is ODD.
Proof :
```Either n or n+1 is even, so their product n(n+1) is even.
Therefore n(n+1)+1 is odd !!
Simple isn't it.```

Sources Revealed

 Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online : Neil Sloane's Integer Sequences One can find the regular numbers of the form n2+(n+1) at %N Central polygonal numbers: n^2 – n + 1 under A002061. The palindromic numbers of the form n2+(n+1) are categorised as follows : %N n^2 + (n+1) is a palindrome under A028413. %N Palindromes of the form n^2 + (n+1) under A028414. If one searches the database for the short sequence {1,3,7,13} you get apositive reply and one of the entries shown is : %N Primes of form n^2 + n + 1 under A002383. %N n^2 + n + 1 is prime under A002384. 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.

I will give a proof that every number of the form n(n – 1) + 1 is also a member of the numbers of the form n^2 + (n+1).
Substitute n with m+1 and we derive :
n(n – 1) + 1 =
(m+1) ( (m+1) – 1) + 1 =
(m+1) (m) + 1 =
m^2 + m + 1 =
m^2 + (m+1) QED

Two infinite expanding patterns are detected :
The first one starts with :
102 + 11 = 101
1002 + 101 = 10101
10002 + 1001 = 1001001
100002 + 10001 = 100010001
The stepvalue of the length of palindromes = + 2 and starts with length 3

The second pattern is less obvious :
11202 + 1121 = 1255521
10102002 + 1010201 = 1020505050201
10010020002 + 1001002001 = 1002005005005002001
10001000200002 + 1000100020001 = 1000200050005000500020001
The stepvalue of the length of palindromes = + 6 and starts with length 7.

And the following phenomenon totally surprised me !
All the 'least significant' terms of the above pattern when multiplied with their reversals produce palindromes :
1121 x 1211 = 1357531
1010201 x 1020101 = 1030507050301
1001002001 x 1002001001 = 1003005007005003001
1000100020001 x 1000200010001 = 1000300050007000500030001

A lot of the palindromes are semiprimes. Is there a logical explanation for this phenomenon ?

```
[9]
F_qs  30.103 =
PRIME

Comm  30103 is the only known multi-digit palindromic prime
of the form sigma(n)/d(n), when n is composite. ( McCranie and Honaker )
sigma(n) = the sum of all the factors of n, including 1 and n itself.
d(n) = the number of factors of n, including 1 and n itself.

30103 is also nice because it comes up for
n = 149645 (an odd number) and once for
n = 179574 (an even number).

n         sigma(n)  d(n)  sigma(n)/d(n)=palprime
149645    180618    6     30103
179574    361236    12    30103

30103 is still the only "known" palprime with the property.

```

The Table

Index NrBasenumberLength
Palindromic Quasi_Over_Squares of form n^2+(n+1)Length

[QM1] Palindromic Quasi_Over_Squares
One can find the regular numbers of the form n2+(n+1) at
%N n^2 + (n+1). under A002061.
The palindromic numbers of the form n2+(n+1) are categorised as follows :
%N n^2 + (n+1) is a palindrome. under A028413.
%N Palindromes of the form n^2 + (n+1). under A028414.
150198.673.328.269.067.804.729.345.53127
39.471.091.365.508.776.518.863.915.551.936.881.567.780.556.319.017.49353
149187.537.335.862.291.320.487.992.17627
35.170.252.342.325.858.219.444.225.952.244.491.285.852.324.325.207.15353
148182.095.674.951.859.210.012.760.63627
33.158.834.836.173.165.706.798.349.094.389.760.756.137.163.843.885.13353
147137.880.643.739.559.537.595.197.13427
19.011.071.918.035.338.707.765.218.881.256.770.783.353.081.917.011.09153
146130.295.095.225.061.266.208.421.63927
16.976.811.839.707.783.173.545.748.184.754.537.138.770.793.811.867.96153
145120.717.789.164.926.294.593.811.68427
14.572.784.620.867.596.289.855.881.918.855.898.269.576.802.648.727.54153
144110.535.170.284.836.050.369.573.19527
12.218.023.869.897.702.580.590.247.574.209.508.520.779.896.832.081.22153
143109.111.610.074.229.017.084.985.18927
11.905.343.452.990.595.131.230.428.682.403.213.159.509.925.434.350.91153
142100.000.000.000.000.000.000.000.00027
10.000.000.000.000.000.000.000.000.100.000.000.000.000.000.000.000.00153
14119.594.295.446.676.361.518.130.38626
383.936.414.051.641.993.745.561.222.165.547.399.146.150.414.639.38351
14019.383.866.944.343.600.449.187.50826
375.734.297.716.016.509.914.016.656.610.419.905.610.617.792.437.57351
13912.825.001.501.380.972.030.499.23526
164.480.663.510.424.186.727.128.565.821.727.681.424.015.366.084.46151
13812.440.854.037.620.973.420.974.95026
154.774.849.185.390.076.751.797.434.797.157.670.093.581.948.477.45151
13711.523.152.676.156.629.123.395.20526
132.783.047.598.015.683.580.806.525.608.085.386.510.895.740.387.23151
13611.443.704.882.753.127.488.830.99426
130.958.381.443.547.771.365.974.505.479.563.177.745.344.183.859.03151
13510.000.000.000.000.000.000.000.00026
100.000.000.000.000.000.000.000.010.000.000.000.000.000.000.000.00151
1341.866.690.185.312.364.850.600.39325
3.484.532.247.941.511.026.607.063.607.066.201.151.497.422.354.84349
1331.203.507.603.075.463.896.117.15925
1.448.430.550.660.448.354.463.702.073.644.538.440.660.550.348.44149
1321.000.000.010.000.000.200.000.00025
1.000.000.020.000.000.500.000.005.000.000.050.000.000.200.000.00149
1311.000.000.000.000.000.000.000.00025
1.000.000.000.000.000.000.000.001.000.000.000.000.000.000.000.00149
130265.455.445.544.554.455.445.43724
70.466.593.569.257.915.890.542.124.509.851.975.296.539.566.40747
129174.868.688.603.991.484.485.62624
30.579.058.254.079.741.275.153.435.157.214.797.045.285.097.50347
128126.050.968.010.335.988.738.22524
15.888.846.536.342.746.771.487.678.417.764.724.363.564.888.85147
127109.068.284.003.373.554.072.41424
Prime!    11.895.890.575.440.551.507.330.303.370.515.504.457.509.859.81147
126107.122.509.064.061.532.741.31024
11.475.231.948.179.945.239.373.537.393.254.997.184.913.257.41147
125100.000.000.000.000.000.000.00024
10.000.000.000.000.000.000.000.100.000.000.000.000.000.000.00147
12419.830.431.895.576.339.981.54323
393.246.029.165.091.432.530.060.035.234.190.561.920.642.39345
12319.311.068.891.689.189.856.98323
372.917.381.739.565.955.495.848.594.559.565.937.183.719.27345
12218.448.546.164.376.797.088.99323
340.348.855.579.141.831.877.656.778.138.141.975.558.843.04345
12117.988.729.019.015.444.317.36623
323.594.371.719.568.349.640.979.046.943.865.917.173.495.32345
12011.442.728.977.039.858.149.00523
130.936.046.441.987.638.522.191.225.836.789.144.640.639.03145
11911.134.188.044.349.820.552.92023
123.970.143.406.942.481.571.868.175.184.249.604.341.079.32145
11810.000.000.000.000.000.000.00023
100.000.000.000.000.000.000.010.000.000.000.000.000.000.00145
1172.646.181.864.462.346.192.40222
7.002.278.459.809.418.714.663.664.178.149.089.548.722.00743
1161.988.871.455.226.161.324.23122
3.955.609.665.413.428.629.029.209.268.243.145.669.065.59343
1151.974.796.881.473.252.797.41322
3.899.822.723.076.484.457.737.377.544.846.703.272.289.98343
1141.866.239.981.123.248.444.39322
3.482.851.667.142.902.710.061.600.172.092.417.661.582.84343
1131.257.643.369.684.401.760.64922
1.581.666.845.311.136.833.493.943.386.311.135.486.661.85143
1121.240.734.862.303.538.673.14922
1.539.422.998.535.381.071.575.751.701.835.358.992.249.35143
1111.042.776.663.690.308.997.80022
1.087.383.170.337.091.795.807.085.971.907.330.713.837.80143
1101.007.779.342.176.626.844.89922
1.015.619.202.517.954.735.095.905.374.597.152.029.165.10143
1091.000.000.100.000.020.000.00022
1.000.000.200.000.050.000.005.000.000.500.000.020.000.00143
1081.000.000.000.000.000.000.00022
1.000.000.000.000.000.000.001.000.000.000.000.000.000.00143
107188.608.385.819.602.861.64821
35.573.123.201.476.169.825.552.896.167.410.232.137.55341
106173.770.124.996.346.340.29821
30.196.056.341.245.831.193.939.113.854.214.365.069.10341
105137.910.980.483.427.117.13421
19.019.438.537.900.215.198.889.151.200.973.583.491.09141
104131.077.218.569.828.232.85421
17.181.237.228.002.523.325.552.332.520.082.273.218.17141
103121.350.673.633.339.893.14021
Prime!    14.725.985.991.265.373.941.814.937.356.219.958.952.74141
102100.000.000.000.000.000.00021
10.000.000.000.000.000.000.100.000.000.000.000.000.00141
10126.545.544.554.455.445.43720
704.665.935.692.579.153.222.351.975.296.539.566.40739
10026.544.554.455.445.544.56220
704.613.371.238.113.910.828.019.311.832.173.316.40739
9910.000.000.000.000.000.00020
100.000.000.000.000.000.010.000.000.000.000.000.00139
981.940.017.649.366.890.18319
3.763.668.479.855.034.063.604.305.589.748.663.67337
971.394.821.668.153.036.60919
1.945.527.485.949.219.781.879.129.495.847.255.49137
961.211.056.769.126.025.58419
1.466.658.498.045.967.635.367.695.408.948.566.64137
951.165.124.977.484.160.63019
1.357.516.213.157.465.816.185.647.513.126.157.53137
941.000.001.000.002.000.00019
1.000.002.000.005.000.005.000.005.000.002.000.00137
931.000.000.000.000.000.00019
1.000.000.000.000.000.001.000.000.000.000.000.00137
92100.000.000.000.000.00018
10.000.000.000.000.000.100.000.000.000.000.00135
9118.516.595.403.268.59317
342.864.305.328.347.606.743.823.503.468.24333
9010.920.545.563.373.68517
119.258.315.401.720.686.027.104.513.852.91133
8910.000.000.000.000.00017
100.000.000.000.000.010.000.000.000.000.00133
882.654.554.455.445.43716
7.046.659.356.925.223.225.296.539.566.40731
872.654.455.445.544.56216
7.046.133.712.381.181.811.832.173.316.40731
861.740.233.076.507.52616
3.028.411.160.570.850.580.750.611.148.20331
851.320.273.129.165.44516
1.743.121.135.596.317.136.955.311.213.47131
841.157.309.182.414.43016
1.339.364.543.700.757.570.073.454.639.33131
831.081.750.100.278.61016
1.170.183.279.452.783.872.549.723.810.71131
821.068.301.518.025.31016
1.141.268.133.415.182.815.143.318.621.41131
811.000.010.000.200.00016
1.000.020.000.500.005.000.050.000.200.00131
801.000.000.000.000.00016
1.000.000.000.000.001.000.000.000.000.00131
79199.104.171.660.06815
39.642.471.172.442.024.427.117.424.69329
78192.656.562.917.31615
37.116.551.235.113.931.153.215.561.17329
77179.734.099.782.36615
32.304.346.624.577.677.542.664.340.32329
76179.298.014.097.15815
32.147.777.859.184.848.195.877.774.12329
75139.549.925.104.60915
19.474.181.596.702.120.769.518.147.49129
74109.959.345.125.37915
12.091.057.580.402.320.408.575.019.02129
73100.000.000.000.00015
10.000.000.000.000.100.000.000.000.00129
7217.760.716.904.12814
315.443.064.948.595.849.460.344.51327
7117.518.506.513.79814
306.898.070.474.000.474.070.898.60327
7010.000.000.000.00014
100.000.000.000.010.000.000.000.00127
691.809.037.732.95813
3.272.617.519.267.629.157.162.72325
681.289.218.480.93913
1.662.084.291.595.951.924.802.66125
671.143.395.132.99413
1.307.352.430.155.510.342.537.03125
661.049.768.613.91013
1.102.014.142.751.572.414.102.01125
65Prime!    1.001.708.756.99913
1.003.420.433.849.483.340.243.00125
641.000.100.020.00013
1.000.200.050.005.000.500.020.00125
631.000.000.000.00013
1.000.000.000.001.000.000.000.00125
62265.455.445.43712
70.466.593.512.421.539.566.40723
61265.445.544.56212
70.461.337.128.082.173.316.40723
60188.582.871.35112
35.563.499.367.176.399.436.55323
59187.343.418.85112
35.097.556.586.968.565.579.05323
58134.717.724.21912
18.148.865.218.881.256.884.18123
57100.000.000.00012
10.000.000.000.100.000.000.00123
5627.509.795.11211
756.788.827.131.728.887.65721
5526.564.008.69711
705.646.558.080.855.646.50721
5418.219.868.63611
331.963.613.131.316.369.13321
5318.099.156.16611
327.579.453.939.354.975.72321
5217.536.911.90111
307.543.279.040.972.345.70321
5112.395.279.27411
153.642.948.292.849.246.35121
5012.369.915.27411
153.014.803.898.308.410.35121
4911.380.055.10411
129.505.654.181.456.505.92121
4811.027.977.09511
121.616.278.818.872.616.12121
4710.798.301.61411
116.603.317.757.713.306.61121
4610.000.000.00011
100.000.000.010.000.000.00121
451.945.980.21610
3.786.839.003.009.386.87319
441.874.554.17610
3.513.953.360.633.593.15319
431.001.002.00010
1.002.005.005.005.002.00119
421.000.000.00010
1.000.000.001.000.000.00119
41179.427.8419
32.194.350.305.349.12317
40100.000.0009
10.000.000.100.000.00117
3927.494.9328
755.971.313.179.55715
3826.545.4378
704.660.252.066.40715
3718.016.0588
324.578.363.875.42315
3612.783.2398
163.411.212.114.36115
35Prime!    12.643.5498
159.859.343.958.95115
3411.842.1848
140.237.333.732.04115
3310.000.0008
100.000.010.000.00115
321.975.3567
3.902.033.302.09313
311.821.6367
3.318.359.538.13313
301.780.7287
3.170.993.990.71313
291.228.4257
1.509.029.209.05113
281.190.9157
1.418.279.728.14113
271.010.2007
1.020.505.050.20113
261.000.0007
1.000.001.000.00113
25196.8136
38.735.553.78311
24Prime!    192.3836
37.011.411.07311
23188.3486
35.475.157.45311
22187.8766
35.297.579.25311
21126.9356
16.112.621.16111
20119.0846
14.181.118.14111
19100.0006
10.000.100.00111
1811.9155
141.979.1419
1710.0005
100.010.0019
161.9614
3.847.4837
151.8764
3.521.2537
141.8364
3.372.7337
131.3554
1.837.3817
121.1844
1.403.0417
111.1204
1.255.5217
101.0004
1.001.0017
9Prime!    1733
Details    Prime!    30.1035
81253
15.7515
71003
10.1015
6272
Prime!    7573
5182
3433
4102
1113
3Prime!    21
Prime!    71
211
Prime!    31
101
11

Contributions

PDG's program searched for palindromes exhaustively up to length 31.

From length 33 on these numbers were discovered by Feng Yuan during [ January 19-25, 2008 ].

[ November 4, 2022 ]
Missing entry [97] was spotted and added by Robert Xiao.

[ November 11, 2022 ]
Missing entries [105], [115] and [116] were added by Robert Xiao.

[ November 13, 2022 ]
Missing entries [124] was added by Robert Xiao.

[ November 14, 2022 ]
Seven new entries [135] up to [141] were discovered by David Griffeath using Rust code by Robert Xiao.

[ June 20, 2023 ]
Nine new entries [142] up to [150] were added by Patrick De Geest using CUDApalin code written by Robert Xiao.

```

```