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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

     PLAIN TEXT QUAPRON 

Palindromic Quasipronic Numbers


flash So far I compiled 104 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 NrBasenumberLength
Palindromic Product of Sequence n(n+2)Length
   
Twofold sequence (n) x (n+2)
Palindromic Quasipronic Numbers
Subsequence of A066619
See also WONplate 191

One can find the regular numbers of the form n(n+2) at
%N n(n+2). under A005563.
The palindromic numbers of the form n(n+2) are categorised as follows :
%N n(n+2) is a palindrome. under A028503.
%N Palindromes of the form n(n+2). under A028504.
104179.952.055.392.509.877.201.98124
32.382.742.239.988.943.174.652.625.647.134.988.993.224.728.32347
10399.999.999.999.999.999.999.99923
9.999.999.999.999.999.999.999.999.999.999.999.999.999.999.99946
10264.825.429.061.399.630.637.04423
4.202.336.252.994.555.796.745.885.476.975.554.992.526.332.02446
10120.566.244.031.577.809.646.31423
422.970.393.566.409.877.340.494.043.778.904.665.393.079.22445
1009.999.999.999.999.999.999.99922
99.999.999.999.999.999.999.999.999.999.999.999.999.999.99944
996.224.291.959.855.223.043.47122
38.741.810.401.518.373.506.999.960.537.381.510.401.814.78344
986.041.636.672.175.825.109.25722
36.501.373.678.579.778.439.933.993.487.797.587.637.310.56344
975.693.758.099.575.300.973.86722
32.418.881.296.479.342.959.977.995.924.397.469.218.881.42344
96999.999.999.999.999.999.99921
999.999.999.999.999.999.999.999.999.999.999.999.999.99942
95554.092.910.028.082.027.05121
307.018.952.943.388.204.176.671.402.883.349.259.810.70342
94290.877.276.146.529.419.15621
84.609.589.778.424.332.617.971.623.342.487.798.590.64841
9399.999.999.999.999.999.99920
9.999.999.999.999.999.999.999.999.999.999.999.999.99940
9290.836.843.323.216.523.52220
8.251.332.104.926.586.308.998.036.856.294.012.331.52840
9189.703.862.578.415.825.65220
8.046.782.961.487.311.140.550.411.137.841.692.876.40840
9089.463.014.832.484.033.24620
8.003.631.022.917.258.135.335.318.527.192.201.363.00840
8973.279.502.387.125.583.30520
5.369.885.470.104.744.062.112.604.474.010.745.889.63540
8864.970.657.724.865.410.71420
4.221.186.365.201.613.466.886.643.161.025.636.811.22440
8724.367.639.093.196.434.91320
593.781.834.976.275.172.828.271.572.679.438.187.39539
869.999.999.999.999.999.99919
99.999.999.999.999.999.999.999.999.999.999.999.99938
859.087.624.822.702.230.72619
82.584.924.918.193.750.455.405.739.181.942.948.52838
846.503.375.380.654.101.68419
42.293.891.341.697.881.988.918.879.614.319.839.22438
833.159.884.456.126.887.92919
9.984.869.776.072.318.331.338.132.706.779.684.89937
823.153.133.622.994.482.34919
9.942.251.644.458.310.353.530.138.544.461.522.49937
81999.999.999.999.999.99918
999.999.999.999.999.999.999.999.999.999.999.99936
80918.101.589.661.512.80618
842.910.528.938.996.839.938.699.839.825.019.24836
79910.786.809.752.004.92218
829.532.612.818.234.809.908.432.818.216.235.92836
78906.394.740.346.503.62618
821.551.425.327.805.729.927.508.723.524.155.12836
77547.914.424.400.753.00118
300.210.216.466.408.476.674.804.664.612.012.00336
7699.999.999.999.999.99917
9.999.999.999.999.999.999.999.999.999.999.99934
7592.087.949.336.643.45617
8.480.190.413.028.212.112.128.203.140.910.84834
7476.110.360.069.816.32317
5.792.786.909.957.091.111.907.599.096.872.97534
7372.909.735.507.526.44317
5.315.829.531.777.462.332.647.771.359.285.13534
7228.296.773.213.909.50217
800.707.374.319.426.343.624.913.473.707.00833
7120.507.633.917.530.25417
420.563.048.895.437.313.734.598.840.365.02433
709.999.999.999.999.99916
99.999.999.999.999.999.999.999.999.999.99932
699.999.711.145.441.49916
99.994.222.992.266.955.966.229.922.249.99932
682.408.106.333.065.82316
5.798.976.111.351.729.271.531.116.798.97531
67999.999.999.999.99915
999.999.999.999.999.999.999.999.999.99930
66243.024.346.003.43515
59.060.832.750.397.779.305.723.806.09529
65185.008.189.318.33715
34.228.030.114.849.994.841.103.082.24329
6499.999.999.999.99914
9.999.999.999.999.999.999.999.999.99928
6364.860.859.637.35414
4.206.931.112.896.666.982.111.396.02428
6224.366.788.929.91314
593.740.402.754.979.457.204.047.39527
6117.556.561.248.20114
308.232.842.861.868.168.248.232.80327
609.999.999.999.99913
99.999.999.999.999.999.999.999.99926
596.035.200.088.90713
36.423.640.113.155.131.104.632.46326
583.153.099.043.04913
9.942.033.575.282.825.753.302.49925
572.967.841.885.46613
8.808.085.457.132.317.545.808.08825
562.967.032.967.03213
8.803.284.627.460.647.264.823.08825
552.939.136.124.28613
8.638.521.157.088.807.511.258.36825
542.839.531.502.89613
8.062.939.155.944.495.519.392.60825
532.050.695.871.49413
4.205.353.557.366.637.553.535.02425
52999.999.999.99912
999.999.999.999.999.999.999.99924
51186.125.268.23712
34.642.615.476.667.451.624.64323
50178.923.207.53112
32.013.514.193.539.141.531.02323
4999.999.999.99911
9.999.999.999.999.999.999.99922
4864.819.595.89411
4.201.580.011.991.100.851.02422
4754.809.426.00111
3.004.073.178.668.713.704.00322
4631.519.410.87911
993.473.262.222.262.374.39921
4524.064.405.17511
579.095.596.474.695.590.97521
4418.511.102.33711
342.660.909.767.909.066.24321
4318.163.818.61711
329.924.306.787.603.429.92321
429.999.999.99910
99.999.999.999.999.999.99920
413.146.471.48910
9.900.282.837.382.820.09919
402.967.032.96610
8.803.284.627.264.823.08819
392.441.052.18510
5.958.735.774.775.378.59519
38999.999.9999
999.999.999.999.999.99918
37185.812.3879
34.526.243.534.262.54317
3699.999.9998
9.999.999.999.999.99916
3593.809.7168
8.800.263.003.620.08816
3431.552.6598
995.570.353.075.59915
3322.765.8958
518.286.020.682.81515
329.999.9997
99.999.999.999.99914
319.200.1567
84.642.888.824.64814
302.967.0327
8.803.284.823.08813
29Prime!    1.868.2877
3.490.500.050.94313
28999.9996
999.999.999.99912
27552.1016
304.816.618.40312
26293.7866
86.310.801.36811
25243.0636
59.080.108.09511
24Prime!    179.3176
32.154.945.12311
23174.7476
30.536.863.50311
22174.6016
30.485.858.40311
2199.9995
9.999.999.99910
2020.5645
422.919.2249
1918.9915
360.696.0639
189.9994
99.999.9998
175.7314
32.855.8238
162.9664
8.803.0887
152.3654
5.597.9557
14Prime!    1.9074
3.640.4637
13Prime!    1.7474
3.055.5037
129993
999.9996
112043
42.0245
10Prime!    1913
36.8635
9992
9.9994
8752
5.7754
7642
4.2244
6Prime!    232
5753
5Prime!    172
3233
491
992
3Prime!    21
81
211
Prime!    31
101
01



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( © All rights reserved ) - Last modified : August 18, 2021.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com