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Palindromic Quasipronic Numbers
of the form n(n+2)
rood n(n+0) rood n(n+1) rood n(n+x) rood
rood n^2+1 rood n^2+x rood n^2–x rood
rood n^2+(n+1) rood n^2+(n+x)


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 of the form x(x+2) can only start or end in one of the following digits : 0, 3, 4, 5, 8 or 9.
3 can only be followed by an even number : 30, 32, 34, 36 or 38
4 can only be followed by 2 : 42
5 can only be followed by an odd number : 51, 53, 55, 57, 59
8 can only be followed by an even number : 80, 82, 84, 86 or 88
9 can only be followed by 9 : 99
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.

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
103179.952.055.392.509.877.201.98124
32.382.742.239.988.943.174.652.625.647.134.988.993.224.728.32347
10299.999.999.999.999.999.999.99923
9.999.999.999.999.999.999.999.999.999.999.999.999.999.999.99946
10164.825.429.061.399.630.637.04423
4.202.336.252.994.555.796.745.885.476.975.554.992.526.332.02446
10020.566.244.031.577.809.646.31423
422.970.393.566.409.877.340.494.043.778.904.665.393.079.22445
999.999.999.999.999.999.999.99922
99.999.999.999.999.999.999.999.999.999.999.999.999.999.99944
986.224.291.959.855.223.043.47122
38.741.810.401.518.373.506.999.960.537.381.510.401.814.78344
976.041.636.672.175.825.109.25722
36.501.373.678.579.778.439.933.993.487.797.587.637.310.56344
965.693.758.099.575.300.973.86722
32.418.881.296.479.342.959.977.995.924.397.469.218.881.42344
95999.999.999.999.999.999.99921
999.999.999.999.999.999.999.999.999.999.999.999.999.99942
94554.092.910.028.082.027.05121
307.018.952.943.388.204.176.671.402.883.349.259.810.70342
93290.877.276.146.529.419.15621
84.609.589.778.424.332.617.971.623.342.487.798.590.64841
9299.999.999.999.999.999.99920
9.999.999.999.999.999.999.999.999.999.999.999.999.99940
9190.836.843.323.216.523.52220
8.251.332.104.926.586.308.998.036.856.294.012.331.52840
9089.703.862.578.415.825.65220
8.046.782.961.487.311.140.550.411.137.841.692.876.40840
8989.463.014.832.484.033.24620
8.003.631.022.917.258.135.335.318.527.192.201.363.00840
8873.279.502.387.125.583.30520
5.369.885.470.104.744.062.112.604.474.010.745.889.63540
8764.970.657.724.865.410.71420
4.221.186.365.201.613.466.886.643.161.025.636.811.22440
8624.367.639.093.196.434.91320
593.781.834.976.275.172.828.271.572.679.438.187.39539
859.999.999.999.999.999.99919
99.999.999.999.999.999.999.999.999.999.999.999.99938
849.087.624.822.702.230.72619
82.584.924.918.193.750.455.405.739.181.942.948.52838
836.503.375.380.654.101.68419
42.293.891.341.697.881.988.918.879.614.319.839.22438
823.159.884.456.126.887.92919
9.984.869.776.072.318.331.338.132.706.779.684.89937
813.153.133.622.994.482.34919
9.942.251.644.458.310.353.530.138.544.461.522.49937
80999.999.999.999.999.99918
999.999.999.999.999.999.999.999.999.999.999.99936
79918.101.589.661.512.80618
842.910.528.938.996.839.938.699.839.825.019.24836
78910.786.809.752.004.92218
829.532.612.818.234.809.908.432.818.216.235.92836
77906.394.740.346.503.62618
821.551.425.327.805.729.927.508.723.524.155.12836
76547.914.424.400.753.00118
300.210.216.466.408.476.674.804.664.612.012.00336
7599.999.999.999.999.99917
9.999.999.999.999.999.999.999.999.999.999.99934
7492.087.949.336.643.45617
8.480.190.413.028.212.112.128.203.140.910.84834
7376.110.360.069.816.32317
5.792.786.909.957.091.111.907.599.096.872.97534
7272.909.735.507.526.44317
5.315.829.531.777.462.332.647.771.359.285.13534
7128.296.773.213.909.50217
800.707.374.319.426.343.624.913.473.707.00833
7020.507.633.917.530.25417
420.563.048.895.437.313.734.598.840.365.02433
699.999.999.999.999.99916
99.999.999.999.999.999.999.999.999.999.99932
689.999.711.145.441.49916
99.994.222.992.266.955.966.229.922.249.99932
672.408.106.333.065.82316
5.798.976.111.351.729.271.531.116.798.97531
66999.999.999.999.99915
999.999.999.999.999.999.999.999.999.99930
65243.024.346.003.43515
59.060.832.750.397.779.305.723.806.09529
64185.008.189.318.33715
34.228.030.114.849.994.841.103.082.24329
6399.999.999.999.99914
9.999.999.999.999.999.999.999.999.99928
6264.860.859.637.35414
4.206.931.112.896.666.982.111.396.02428
6124.366.788.929.91314
593.740.402.754.979.457.204.047.39527
6017.556.561.248.20114
308.232.842.861.868.168.248.232.80327
599.999.999.999.99913
99.999.999.999.999.999.999.999.99926
586.035.200.088.90713
36.423.640.113.155.131.104.632.46326
573.153.099.043.04913
9.942.033.575.282.825.753.302.49925
562.967.841.885.46613
8.808.085.457.132.317.545.808.08825
552.967.032.967.03213
8.803.284.627.460.647.264.823.08825
542.939.136.124.28613
8.638.521.157.088.807.511.258.36825
532.839.531.502.89613
8.062.939.155.944.495.519.392.60825
522.050.695.871.49413
4.205.353.557.366.637.553.535.02425
51999.999.999.99912
999.999.999.999.999.999.999.99924
50186.125.268.23712
34.642.615.476.667.451.624.64323
49178.923.207.53112
32.013.514.193.539.141.531.02323
4899.999.999.99911
9.999.999.999.999.999.999.99922
4764.819.595.89411
4.201.580.011.991.100.851.02422
4654.809.426.00111
3.004.073.178.668.713.704.00322
4531.519.410.87911
993.473.262.222.262.374.39921
4424.064.405.17511
579.095.596.474.695.590.97521
4318.511.102.33711
342.660.909.767.909.066.24321
4218.163.818.61711
329.924.306.787.603.429.92321
419.999.999.99910
99.999.999.999.999.999.99920
403.146.471.48910
9.900.282.837.382.820.09919
392.967.032.96610
8.803.284.627.264.823.08819
382.441.052.18510
5.958.735.774.775.378.59519
37999.999.9999
999.999.999.999.999.99918
36185.812.3879
34.526.243.534.262.54317
3599.999.9998
9.999.999.999.999.99916
3493.809.7168
8.800.263.003.620.08816
3331.552.6598
995.570.353.075.59915
3222.765.8958
518.286.020.682.81515
319.999.9997
99.999.999.999.99914
309.200.1567
84.642.888.824.64814
292.967.0327
8.803.284.823.08813
28Prime!    1.868.2877
3.490.500.050.94313
27999.9996
999.999.999.99912
26552.1016
304.816.618.40312
25293.7866
86.310.801.36811
24243.0636
59.080.108.09511
23Prime!    179.3176
32.154.945.12311
22174.7476
30.536.863.50311
21174.6016
30.485.858.40311
2099.9995
9.999.999.99910
1920.5645
422.919.2249
1818.9915
360.696.0639
179.9994
99.999.9998
165.7314
32.855.8238
152.9664
8.803.0887
142.3654
5.597.9557
13Prime!    1.9074
3.640.4637
12Prime!    1.7474
3.055.5037
119993
999.9996
102043
42.0245
9Prime!    1913
36.8635
8992
9.9994
7752
5.7754
6642
4.2244
5Prime!    232
5753
4Prime!    172
3233
391
992
2Prime!    21
81
111
31



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( © All rights reserved ) - Last modified : January 14, 2021.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com