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Palindromic Quasipronic Numbers
of the form n(n+2)
rood n(n+1) rood n(n+x) 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 Products of Two Sequential Integers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

base x ( base + n )
In you are interested in case n = 1 then visit this page about palindromic pronic numbers !

In case of n = 2 note that the formula n(n+2) can be written also as n2+2n.
Its always easier to refer something by name so I'll baptise these numbers Palindromic Quasipronic Numbers.
A repetitive infinite palindromic pattern can be made in the following manner :

9 x 11 = 99
99 x 101 = 9999
999 x 1001 = 999999
9999 x 10001 = 99999999


Palindromic Quasipronic Numbers


flash So far I compiled 103 Palindromic Quasipronic Numbers.
Here is the largest non trivial one that Feng Yuan discovered on [ January 15, 2008 ].

This nonpalindromic basenumber
179.952.055.392.509.877.201.981

has 24 digits
yielding the following palindromic quasipronic number
32.382.742.239.988.943.174.652.625.647.134.988.993.224.728.323
with a length of 47 digits.


Palindromic Quasipronic Numbers can only end in one of the following digits : 3, 4, 5, 8 or 9.

Warut Roonguthai (email) from Bangkok Thailand informed me that every (palindromic) number
of the form n(n+2) is also of the form n^2–1.
" It's just one step away from being a palindromic square. And that is why I think that it is interesting
to investigate
palindromes of the form n^2+1 , another near miss, as well. "

Proof that numbers of the form n^2–1 equal n(n+2) by substituting n with m+1 :

n^2 – 1 =
(m + 1)^2 – 1 =
m^2 + 2m + 1 – 1 =
m^2 + 2m =
m(m + 2)  QED


A finite palindromic pattern hides in the list in the following manner :

2 x 4 = 8
2966 x 2968 = 8803088
2967032 x 2967034 = 8803284823088
2967032966 x 2967032968 = 8803284627264823088
2967032967032 x 2967032967034 = 8803284627460647264823088

Note that the lengths of the resulting palindromes increases with steps of 6 !


Sources Revealed


Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
The regular quasipronic numbers are categorised as follows :
%N n(n+2) under A005563 .
Check out the following two entries about Palindromic Quasipronic Numbers
%N n(n+2) is a palindrome under A028503.
%N Palindromes of form n(n+2) under A028504.
%N Both n and its reverse are one less than a square under A066619.
Click here to view some the author's [P. De Geest] entries to the table.
Click here to view some entries to the table about palindromes.


The Table

My program completed the search upto length 26 inclusive.

From length 27 on these quasipronic numbers were submitted by Feng Yuan on [ January 15, 2008 ].




Index NrInfo BasenumberLength
Palindromic Product of Sequence n(n+2)Length
   
Twofold sequence (n) x (n+2)
Palindromic Quasipronic Numbers
Subsequence of A066619
See also WONplate 191
103 179.952.055.392.509.877.201.98124
32.382.742.239.988.943.174.652.625.647.134.988.993.224.728.32347
102 99.999.999.999.999.999.999.99923
9.999.999.999.999.999.999.999.999.999.999.999.999.999.999.99946
101 64.825.429.061.399.630.637.04423
4.202.336.252.994.555.796.745.885.476.975.554.992.526.332.02446
100 20.566.244.031.577.809.646.31423
422.970.393.566.409.877.340.494.043.778.904.665.393.079.22445
99 9.999.999.999.999.999.999.99922
99.999.999.999.999.999.999.999.999.999.999.999.999.999.99944
98 6.224.291.959.855.223.043.47122
38.741.810.401.518.373.506.999.960.537.381.510.401.814.78344
97 6.041.636.672.175.825.109.25722
36.501.373.678.579.778.439.933.993.487.797.587.637.310.56344
96 5.693.758.099.575.300.973.86722
32.418.881.296.479.342.959.977.995.924.397.469.218.881.42344
95 999.999.999.999.999.999.99921
999.999.999.999.999.999.999.999.999.999.999.999.999.99942
94 554.092.910.028.082.027.05121
307.018.952.943.388.204.176.671.402.883.349.259.810.70342
93 290.877.276.146.529.419.15621
84.609.589.778.424.332.617.971.623.342.487.798.590.64841
92 99.999.999.999.999.999.99920
9.999.999.999.999.999.999.999.999.999.999.999.999.99940
91 90.836.843.323.216.523.52220
8.251.332.104.926.586.308.998.036.856.294.012.331.52840
90 89.703.862.578.415.825.65220
8.046.782.961.487.311.140.550.411.137.841.692.876.40840
89 89.463.014.832.484.033.24620
8.003.631.022.917.258.135.335.318.527.192.201.363.00840
88 73.279.502.387.125.583.30520
5.369.885.470.104.744.062.112.604.474.010.745.889.63540
87 64.970.657.724.865.410.71420
4.221.186.365.201.613.466.886.643.161.025.636.811.22440
86 24.367.639.093.196.434.91320
593.781.834.976.275.172.828.271.572.679.438.187.39539
85 9.999.999.999.999.999.99919
99.999.999.999.999.999.999.999.999.999.999.999.99938
84 9.087.624.822.702.230.72619
82.584.924.918.193.750.455.405.739.181.942.948.52838
83 6.503.375.380.654.101.68419
42.293.891.341.697.881.988.918.879.614.319.839.22438
82 3.159.884.456.126.887.92919
9.984.869.776.072.318.331.338.132.706.779.684.89937
81 3.153.133.622.994.482.34919
9.942.251.644.458.310.353.530.138.544.461.522.49937
80 999.999.999.999.999.99918
999.999.999.999.999.999.999.999.999.999.999.99936
79 918.101.589.661.512.80618
842.910.528.938.996.839.938.699.839.825.019.24836
78 910.786.809.752.004.92218
829.532.612.818.234.809.908.432.818.216.235.92836
77 906.394.740.346.503.62618
821.551.425.327.805.729.927.508.723.524.155.12836
76 547.914.424.400.753.00118
300.210.216.466.408.476.674.804.664.612.012.00336
75 99.999.999.999.999.99917
9.999.999.999.999.999.999.999.999.999.999.99934
74 92.087.949.336.643.45617
8.480.190.413.028.212.112.128.203.140.910.84834
73 76.110.360.069.816.32317
5.792.786.909.957.091.111.907.599.096.872.97534
72 72.909.735.507.526.44317
5.315.829.531.777.462.332.647.771.359.285.13534
71 28.296.773.213.909.50217
800.707.374.319.426.343.624.913.473.707.00833
70 20.507.633.917.530.25417
420.563.048.895.437.313.734.598.840.365.02433
69 9.999.999.999.999.99916
99.999.999.999.999.999.999.999.999.999.99932
68 9.999.711.145.441.49916
99.994.222.992.266.955.966.229.922.249.99932
67 2.408.106.333.065.82316
5.798.976.111.351.729.271.531.116.798.97531
66 999.999.999.999.99915
999.999.999.999.999.999.999.999.999.99930
65 243.024.346.003.43515
59.060.832.750.397.779.305.723.806.09529
64 185.008.189.318.33715
34.228.030.114.849.994.841.103.082.24329
63 99.999.999.999.99914
9.999.999.999.999.999.999.999.999.99928
62 64.860.859.637.35414
4.206.931.112.896.666.982.111.396.02428
61 24.366.788.929.91314
593.740.402.754.979.457.204.047.39527
60 17.556.561.248.20114
308.232.842.861.868.168.248.232.80327
59 9.999.999.999.99913
99.999.999.999.999.999.999.999.99926
58 6.035.200.088.90713
36.423.640.113.155.131.104.632.46326
57 3.153.099.043.04913
9.942.033.575.282.825.753.302.49925
56 2.967.841.885.46613
8.808.085.457.132.317.545.808.08825
55 2.967.032.967.03213
8.803.284.627.460.647.264.823.08825
54 2.939.136.124.28613
8.638.521.157.088.807.511.258.36825
53 2.839.531.502.89613
8.062.939.155.944.495.519.392.60825
52 2.050.695.871.49413
4.205.353.557.366.637.553.535.02425
51 999.999.999.99912
999.999.999.999.999.999.999.99924
50 186.125.268.23712
34.642.615.476.667.451.624.64323
49 178.923.207.53112
32.013.514.193.539.141.531.02323
48 99.999.999.99911
9.999.999.999.999.999.999.99922
47 64.819.595.89411
4.201.580.011.991.100.851.02422
46 54.809.426.00111
3.004.073.178.668.713.704.00322
45 31.519.410.87911
993.473.262.222.262.374.39921
44 24.064.405.17511
579.095.596.474.695.590.97521
43 18.511.102.33711
342.660.909.767.909.066.24321
42 18.163.818.61711
329.924.306.787.603.429.92321
41 9.999.999.99910
99.999.999.999.999.999.99920
40 3.146.471.48910
9.900.282.837.382.820.09919
39 2.967.032.96610
8.803.284.627.264.823.08819
38 2.441.052.18510
5.958.735.774.775.378.59519
37 999.999.9999
999.999.999.999.999.99918
36 185.812.3879
34.526.243.534.262.54317
35 99.999.9998
9.999.999.999.999.99916
34 93.809.7168
8.800.263.003.620.08816
33 31.552.6598
995.570.353.075.59915
32 22.765.8958
518.286.020.682.81515
31 9.999.9997
99.999.999.999.99914
30 9.200.1567
84.642.888.824.64814
29 2.967.0327
8.803.284.823.08813
28 1.868.2877
3.490.500.050.94313
27 999.999 x 1.000.0016-7
999.999.999.99912
26 552.101 x 552.1036
304.816.618.40312
25 293.786 x 293.7886
86.310.801.36811
24 243.063 x 243.0656
59.080.108.09511
23 179.317 x 179.3196
32.154.945.12311
22 174.747 x 174.7496
30.536.863.50311
21 174.601 x 174.6036
30.485.858.40311
20 99.999 x 100.0015-6
9.999.999.99910
19 20.564 x 20.5665
422.919.2249
18 18.991 x 18.9935
360.696.0639
17 9.999 x 10.0014-5
99.999.9998
16 5.731 x 5.7334
32.855.8238
15 2.966 x 2.9684
8.803.0887
14 2.365 x 2.3674
5.597.9557
13 1.907 x 1.9094
3.640.4637
12 1.747 x 1.7494
3.055.5037
11 999 x 1.0013-4
999.9996
10 204 x 2063
42.0245
9 191 x 1933
36.8635
8 99 x 1012-3
9.9994
7 75 x 772
5.7754
6 64 x 662
4.2244
5 23 x 252
5753
4 17 x 192
3233
3 9 x 111-2
992
2 2 x 41
81
1 1 x 31
31



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( © All rights reserved ) - Last modified : October 27, 2016.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com