HOME plateWON | World!OfNumbers Palindromic Quasipronicsof the form n(n+x) n(n+1) n(n+2)

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

 base ( base + X )

Palindromic Quasipronics

An interesting infinite palindromic pattern hides in the list for case X = 3

28 x 31 = 868
298 x 301 = 89698
2998 x 3001 = 8996998
29998 x 30001 = 899969998
299998 x 300001 = 89999699998
2999998 x 3000001 = 8999996999998
29999998 x 30000001 = 899999969999998

A very nice but finite palindromic pattern hides in the list for case X = 5

202 x 207 = 41814
20402 x 20407 = 416343614
2040402 x 2040407 = 4163250523614

A very nice infinite twofold palindromic pattern hides in the list for case X = 7

19 x 26 = 494
219 x 226 = 49494
2109 x 2116 = 4462644
21009 x 21016 = 441525144
210009 x 210016 = 44105250144
2100009 x 2100016 = 4410052500144
21000009 x 21000016 = 441000525000144
Note that if you add n to n+7 and subtract 1 from the above pattern
you get also a palindromic sequence !
19 + 261 = 44
219 + 2261 = 444
2109 + 21161 = 4224
21009 + 210161 = 42024
210009 + 2100161 = 420024
2100009 + 21000161 = 4200024
21000009 + 210000161 = 42000024
And then there is this astonishing infinite pattern where the multiplier n+7 is palindromic itself !
902 x 909 = 819918
9002 x 9009 = 81099018
90002 x 90009 = 8100990018
900002 x 900009 = 810009900018
90000002 x 90000009 = 81000099000018

Another beautiful infinite palindromic pattern hides in the list for case X = 8

66 x 74 = 4884
696 x 704 = 489984
6996 x 7004 = 48999984
69996 x 70004 = 4899999984
699996 x 700004 = 489999999984
6999996 x 7000004 = 48999999999984

Can you detect more hidden patterns in these sequences ?

The Table

Palindromic Quasipronics of form n(n+X)

Case X = 0   [Scanned exhaustively upto length 32]
See in-depth webpage about Palindromic Square Numbers

Case X = 1   [Scanned exhaustively upto length 34]
See in-depth webpage about Palindromic Pronic Numbers

Case X = 2   [Scanned exhaustively upto length 24]
See in-depth webpage about Palindromic Quasipronic Numbers

Case X = 3

One can find the regular numbers of the form n(n+3) at
%N n(n+3). under A028552.
The palindromic numbers of the form n(n+3) are categorised as follows :
%N n(n+3) is a palindrome. under A028553.
%N Palindromes of the form n(n+3). under A028554.
332.999.999.99810
8.999.999.996.999.999.99819
322.946.920.84410
8.684.342.469.642.434.86819
312.134.473.70610
4.555.978.008.008.795.55419
30299.999.9989
89.999.999.699.999.99817
29294.174.6699
86.538.736.763.783.56817
28294.127.3289
86.510.885.958.801.56817
27289.053.6839
83.552.032.523.025.53817
26212.325.2069
45.081.993.739.918.05417
2529.999.9988
899.999.969.999.99815
2428.862.8838
833.066.101.660.33815
2321.341.6918
455.467.838.764.55415
229.443.4237
89.178.266.287.19814
212.999.9987
8.999.996.999.99813
202.861.4197
8.187.727.277.81813
192.122.2067
4.503.764.673.05413
18636.7766
405.485.584.50412
17299.9986
89.999.699.99811
16Prime!          212.6716
45.229.592.25411
15212.2066
45.032.023.05411
1463.7015
4.058.008.50410
1329.9985
899.969.9989
1229.3695
862.626.2689
1128.8145
830.333.0389
102.9984
8.996.9987
92.1264
4.526.2547
86713
452.2546
72983
89.6985
6Prime!          2113
45.1545
5882
8.0084
4662
4.5544
3282
8683
281
882
111
41
001
01

Case X = 4
One can find the regular numbers of the form n(n+4) at
%N n(n+4). [= m^2 – 4] under A028347.
The palindromic numbers of the form n(n+4) are categorised as follows :
%N n(n+4) is a palindrome. under A028555.
%N Palindromes of the form n(n+4). under A028556.
391.455.445.54410
2.118.321.737.371.238.11219
381.106.902.51310
1.225.233.177.713.325.22119
37486.849.8549
237.022.782.287.220.73218
36282.442.3479
79.773.680.508.637.79717
35277.945.1579
77.253.511.411.535.27717
34223.674.4959
50.030.280.608.203.00517
33Prime!          110.651.0639
12.243.658.185.634.22117
3285.559.3278
7.320.398.778.930.23716
3114.554.4528
211.832.131.238.11215
308.308.3887
69.029.344.392.09614
297.395.0917
54.687.400.478.64514
285.040.8827
25.410.511.501.45214
273.504.0837
12.278.611.687.22114
262.705.6697
7.320.655.560.23713
251.659.0227
2.752.360.632.57213
24281.3496
79.158.385.19711
23266.7376
71.149.694.11711
22262.8086
69.069.096.09611
21145.5446
21.183.638.11211
2073.4915
5.401.221.04510
1923.2555
540.888.0459
1811.2235
126.000.6219
1711.0635
122.434.2219
168.4594
71.588.5178
158.3384
69.555.5968
145.4124
29.311.3928
131.5824
2.509.0527
121.4524
2.114.1127
11Prime!          1.1234
1.265.6217
105243
276.6726
9Prime!          2693
73.4375
82353
56.1655
71443
21.3125
6442
2.1124
5332
1.2214
4212
5253
3142
2523
2Prime!          71
772
111
51
001
01

Case X = 5
One can find the regular numbers of the form n(n+5) at
%N n(n+5). under A028557.
The palindromic numbers of the form n(n+5) are categorised as follows :
%N n(n+5) is a palindrome. under A028558.
%N Palindromes of the form n(n+5). under A028559.
238.161.664.18110
66.612.762.244.226.721.66620
22249.281.9869
62.141.509.790.514.12617
21211.313.1489
44.653.247.574.235.64417
2079.585.8918
6.333.914.444.193.33616
196.887.0787
47.431.877.813.47414
182.233.0977
4.986.733.376.89413
172.187.1427
4.783.601.063.87413
162.040.4027
4.163.250.523.61413
15207.0676
42.877.777.82411
1482.5665
6.817.557.18610
1364.4935
4.159.669.51410
1225.1165
630.939.0399
1120.4025
416.343.6149
10Prime!          2.2074
4.881.8847
92.1784
4.754.5747
82.0674
4.282.8247
78143
666.6666
62033
42.2245
52023
41.8145
4242
6963
3182
4143
261
662
111
61
001
01

Case X = 6
One can find the regular numbers of the form n(n+6) at
%N n(n+6). under A028560.
The palindromic numbers of the form n(n+6) are categorised as follows :
%N n(n+6) is a palindrome. under A028561.
%N Palindromes of the form n(n+6). under A028562.
301.414.071.39710
1.999.597.924.297.959.99119
29782.357.2629
612.082.890.098.280.21618
28440.021.4179
193.618.850.058.816.39118
27272.515.5139
74.264.706.460.746.24717
26235.769.7559
55.587.378.787.378.55517
25235.680.7559
55.545.419.691.454.55517
2488.837.6118
7.892.121.661.212.98716
2384.890.5038
7.206.398.008.936.02716
2273.114.0358
5.345.662.552.665.43516
2123.975.4758
574.823.545.328.47515
2023.887.4198
570.608.929.806.07515
1914.529.4868
211.106.050.601.11215
187.560.0697
57.154.688.645.17514
177.153.6797
51.175.166.157.11514
165.401.3967
29.175.111.157.19214
152.436.0997
5.934.592.954.39513
142.273.9157
5.170.703.070.71513
131.731.7467
2.998.954.598.99213
121.378.5077
1.900.289.820.09113
11877.1736
769.437.734.96712
10486.6186
236.799.997.63212
98.5414
72.999.9278
8Prime!          2.7314
7.474.7477
7Prime!          8633
749.9476
67153
515.5156
52733
76.1675
4Prime!          1373
19.5915
3222
6163
2Prime!          51
552
111
71
001
01

Case X = 7
One can find the regular numbers of the form n(n+7) at
%N n(n+7). under A028563.
The palindromic numbers of the form n(n+7) are categorised as follows :
%N n(n+7) is a palindrome. under A028564.
%N Palindromes of the form n(n+7). under A028565.
39900.000.0029
810.000.009.900.000.01818
38703.210.9749
494.505.678.876.505.49418
37665.012.0449
442.241.023.320.142.24418
36221.990.2149
49.279.656.665.697.29417
35210.000.0099
44.100.005.250.000.14417
3490.000.0028
8.100.000.990.000.01816
3329.857.8868
891.493.565.394.19815
3229.802.8978
888.212.878.212.88815
3128.867.7228
833.345.575.543.33815
3021.155.3448
447.548.727.845.74415
2921.152.8248
447.442.111.244.74415
2821.117.3048
445.940.676.049.54415
2721.037.9248
442.594.393.495.24415
2621.000.0098
441.000.525.000.14415
259.000.0027
81.000.099.000.01814
248.972.8767
80.512.566.521.50814
232.219.7797
4.927.434.347.29413
222.109.6647
4.450.696.960.54413
212.100.0097
4.410.052.500.14413
20942.9976
889.249.942.98812
19900.0026
810.009.900.01812
18669.0246
447.597.795.74412
17292.9676
85.831.713.85811
16223.1146
49.781.418.79411
15210.0096
44.105.250.14411
1490.0025
8.100.990.01810
1329.6965
882.060.2889
1221.0095
441.525.1449
119.0024
81.099.0188
102.9824
8.913.1987
92.9724
8.853.5887
82.1094
4.462.6447
79023
819.9186
66643
445.5446
52193
49.4945
4262
8583
3Prime!          192
4943
241
442
111
81
001
01

Case X = 8
One can find the regular numbers of the form n(n+8) at
%N n(n+8). under A028566.
The palindromic numbers of the form n(n+8) are categorised as follows :
%N n(n+8) is a palindrome. under A028567.
%N Palindromes of the form n(n+8). under A028568.
49699.999.9969
489.999.999.999.999.98418
48695.334.3769
483.489.900.009.984.38418
47287.876.0589
82.872.627.072.627.82817
46225.207.1859
50.718.277.977.281.70517
45182.901.1039
33.452.814.941.825.43317
4491.842.9388
8.435.125.995.215.34816
4391.050.7848
8.290.245.995.420.92816
4269.999.9968
4.899.999.999.999.98416
4169.692.7368
4.857.078.008.707.58416
4059.318.0098
3.518.626.666.268.15316
3957.524.7538
3.309.097.667.909.03316
3823.352.3278
545.331.363.133.54515
37Prime!          19.782.1998
391.335.555.533.19315
36Prime!          17.869.1698
319.307.343.703.91315
3517.646.6198
311.403.303.304.11315
347.243.2257
52.464.366.346.42514
336.999.9967
48.999.999.999.98414
322.281.8177
5.206.707.076.02513
31Prime Curios!          1.934.0637
3.740.615.160.47313
30896.4786
803.679.976.30812
29699.9966
489.999.999.98412
28597.8836
357.468.864.75312
27298.1446
88.892.229.88811
26283.7786
88.892.229.88811
25241.0976
58.129.692.18511
24228.1756
52.065.656.02511
23188.5836
35.565.056.55311
2293.8445
8.807.447.08810
2169.9965
4.899.999.98410
2057.4895
3.305.445.03310
1930.0315
902.101.2099
1828.3145
801.909.1089
179.3944
88.322.3888
166.9964
48.999.9848
152.8284
8.020.2087
142.2354
5.013.1057
139243
861.1686
127073
505.5056
116963
489.9846
102943
88.7885
92843
82.9285
82253
52.4255
72163
48.3845
6Prime!          1733
31.3135
5912
9.0094
4882
8.4484
3662
4.8844
2Prime!          31
332
111
91
001
01

Case X = 9
One can find the regular numbers of the form n(n+9) at
%N n(n+9). under A028569.
The palindromic numbers of the form n(n+9) are categorised as follows :
%N n(n+9) is a palindrome. under A028570.
%N Palindromes of the form n(n+9). under A028571.
3915.114.678.51411
228.453.506.717.605.354.82221
381.665.597.36410
2.774.214.593.954.124.77219
371.632.565.35710
2.665.269.659.569.625.66219
36Prime!          1.534.664.34710
2.355.194.671.764.915.53219
351.458.904.65910
2.128.402.817.182.048.21219
341.417.320.72910
2.008.798.061.608.978.00219
33514.668.9579
264.884.139.931.488.46218
32497.665.8749
247.671.326.623.176.74218
31164.399.7279
27.027.271.717.272.07217
30151.698.4729
23.012.427.772.421.03217
2954.072.5248
2.923.838.338.383.29216
2852.030.3528
2.707.157.997.517.07216
2749.407.0178
2.441.053.773.501.44216
2648.675.3198
2.369.287.117.829.63216
2526.216.7538
687.318.373.813.78615
2426.211.4388
687.039.717.930.78615
2326.132.5788
682.911.868.119.28615
2225.234.0838
636.759.171.957.63615
2115.336.6448
235.212.787.212.53215
2015.019.6148
225.588.939.885.52215
1914.790.5328
218.759.969.957.81215
1814.674.1848
215.331.808.133.51215
171.547.1477
2.393.677.763.93213
16487.6196
237.776.677.73212
15160.1876
25.661.316.65211
1416.9975
289.050.9829
1315.9045
253.080.3529
125.3924
29.122.1928
111.6944
2.884.8827
101.6644
2.783.8727
91.6394
2.701.0727
82583
68.8865
72483
63.7365
6Prime!          1673
29.3925
5Prime!          1573
26.0625
4Prime!          1373
20.0025
3442
2.3324
2122
2523
1Prime!          21
222
001
01
```

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Further Topics Revealed

 Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online : Neil Sloane's Integer Sequences That case X=2 with formula n(n+2) equalling n2–1 was already known to me. I was quite surprised to find an entry for case X = 4. %N n^2 – 4 denominators of wavelengths of hydrogen lines. under A028347. Numbers of the form n2–Y. A while ago I submitted numbers of the form n(n+X) to Sloane's table. I found that when X equals an even number there is a corresponding formula n2–Y. whereby Y = (X/2)2 or the square of the natural numbers. n(n+0) = n2–0 (trivial case) n(n+2) = n2–1 n(n+4) = n2–4 n(n+6) = n2–9 n(n+8) = n2–16 n(n+10) = n2–25 The variables at the left are the even numbers (0,2,4,6,8,10,...). The variables at the right are the squares of the natural numbers (0,1,4,9,16,25,...). Obviously all these numbers are composite. The intermediate Y variables can deliver primes. Here's a resumée : %N n^2 – 2. under A008865 %N n^2 – 2 is prime. under A028870 %N Primes of the form n^2 – 2. under A028871 %N n^2 – 3. under A028872 %N n^2 – 3 is prime. under A028873 %N Primes of the form n^2 – 3. under A028874 %N n^2 – 5. under A028875 %N n^2 – 5 is prime. under A028876 %N Primes of the form n^2 – 5. under A028877 %N n^2 – 6. under A028878 %N n^2 – 6 is prime. under A028879 %N Primes of the form n^2 – 6. under A028880 %N n^2 – 7. under A028881 %N n^2 – 7 is prime. under A028882 %N Primes of the form n^2 – 7. under A028883 %N n^2 – 8. under A028884 %N n^2 – 8 is prime. under A028885 %N Primes of the form n^2 – 8. under A028886 This last sequence is a subset of : %N Divisible only by primes congruent to 1 mod 8. under A004625 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.

Contributions
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