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Palindromic Quasi_Over_Squares
of the form n^2+(n+X)
rood n^2+1 rood n^2+(n+1) 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 : 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 + X )


Palindromic Quasi_Over_Squares

Some people regard 1997 as a bad year !

So does Vincent Prosper because he figured out that after applying the "add & reverse" procedure to 1997,
this yearnumber never transformed into a palindrome. Follow this link to arrive at his website.
1997: A Bad Year (broken link!)

As for me, 1997 is a lucky year because of the following relations with the number of the beast 666 and other various palindromic numbers :

14132 + ( 1413 + 9 ) = 1997991 or from the right 1997991.
Note that 1413 = 666 + 747

300032 + ( 30003 + 1997 ) = 900212009.
Note that 1997 = 666 + 1331
and moreover 1331 = 113 is a palindromic cube !


An infinite palindromic pattern resides in the list for case n^2 + (n + 2)

212 + 23 = 464
2012 + 203 = 40604
20012 + 2003 = 4006004
200012 + 20003 = 400060004
2000012 + 200003 = 40000600004
20000012 + 2000003 = 4000006000004
200000012 + 20000003 = 400000060000004

An infinite palindromic pattern hides in the list for case n^2 + (n + 4)

202 + 24 = 424
2002 + 204 = 40204
20002 + 2004 = 4002004
200002 + 20004 = 400020004
2000002 + 200004 = 40000200004
20000002 + 2000004 = 4000002000004
200000002 + 20000004 = 400000020000004

A very nice but finite palindromic pattern hides in the list for case n^2 + (n + 6)

282 + 34 = 818
2882 + 294 = 83238
28882 + 2894 = 8343438
288882 + 28894 = 834545438
2888882 + 288894 = 83456565438
28888882 + 2888894 = 8345676765438
288888882 + 28888894 = 834567878765438
2888888882 + 288888894 = 83456789898765438
alas
28888888882 + 2888888894 = 8345679010098765438
is no longer palindromic !

An infinite palindromic pattern hides in the list for case n^2 + (n + 7)

312 + 38 = 999
3012 + 308 = 90909
30012 + 3008 = 9009009
300012 + 30008 = 900090009
3000012 + 300008 = 90000900009
30000012 + 3000008 = 9000009000009
300000012 + 30000008 = 900000090000009

An infinite palindromic pattern hides in the list for case n^2 + (n + 9)

302 + 39 = 939
3002 + 309 = 90309
30002 + 3009 = 9003009
300002 + 30009 = 900030009
3000002 + 300009 = 90000300009
30000002 + 3000009 = 9000003000009
300000002 + 30000009 = 900000030000009


The Table


Index Nr BasenumberLength
Palindromic Quasi_Over_Squares of form n^2+(n+X)Length
up down
Case X = 0
See in-depth webpage about Palindromic Pronic Numbers
up down
Case X = 1
See in-depth webpage about Palindromic Quasi-Over-Squares
up down
Case X = 2
One can find the regular numbers of the form n2+(n+2) at
%N n^2+n+2. under A014206.
The palindromic numbers of the form n2+(n+2) are categorised as follows :
%N n^2 + (n+2) is a palindrome. under A027712.
%N Palindromes of the form n^2 + (n+2). under A027713.
492.000.000.00110
4.000.000.006.000.000.00419
481.553.608.45910
2.413.699.245.429.963.14219
471.434.631.49910
2.058.167.539.357.618.50219
46528.342.9299
279.146.251.152.641.97218
45482.441.6699
232.749.964.469.947.23218
44223.314.9569
49.869.569.796.596.89417
43213.934.6239
45.768.023.132.086.75417
42200.000.0019
40.000.000.600.000.00417
41161.280.5399
26.011.412.421.411.06217
4092.610.8528
8.576.770.000.776.75816
3948.829.9308
2.384.362.112.634.83216
3829.287.0528
857.731.444.137.75815
3722.012.0388
484.529.838.925.48415
3621.855.1088
477.645.767.546.77415
3521.147.9068
447.233.949.332.74415
3420.000.0018
400.000.060.000.00415
3317.101.8098
292.471.888.174.29215
3214.842.9958
220.314.515.413.02215
312.000.0017
4.000.006.000.00413
301.651.7547
2.728.292.928.27213
291.620.6607
2.626.540.456.26213
28501.4746
251.476.674.15212
27Prime!          219.0836
47.997.579.97411
26200.0016
40.000.600.00411
2589.7775
8.059.999.50810
2428.3775
805.282.5089
2320.9935
440.727.0449
2220.0015
400.060.0049
2116.7805
281.585.1829
202.1834
4.767.6747
192.1064
4.437.3447
182.0014
4.006.0047
171.6544
2.737.3727
161.5304
2.342.4327
151.4244
2.029.2029
148973
805.5086
13Prime!          4793
229.9226
122923
85.5585
112033
41.4145
102013
40.6045
9922
8.5584
8502
2.5524
7212
4643
6152
2423
5142
2123
461
442
341
222
2Prime!          21
81
111
41
001
21
up down
Case X = 3
One can find the regular numbers of the form n2+(n+3) at
%N n^2 + (n+3). under A027688.
The palindromic numbers of the form n2+(n+3) are categorised as follows :
%N n^2 + (n+3) is a palindrome. under A027714.
%N Palindromes of the form n^2 + (n+3). under A027715.
362.315.681.86310
5.362.382.492.942.832.63519
35727.913.4339
529.857.966.669.758.92518
34726.714.7419
528.114.315.513.411.82518
33589.167.8599
347.118.766.667.811.74318
32549.199.5249
301.620.117.711.026.10318
31Prime!          309.230.2879
95.623.370.707.332.65917
30230.098.5669
52.945.350.305.354.92517
29177.865.3859
31.636.095.359.063.61317
2876.675.1868
5.879.084.224.809.78516
2762.127.1808
3.859.786.556.879.58316
2630.909.0928
955.371.999.173.55915
2518.886.0258
356.681.959.186.65315
2417.605.7248
309.961.535.169.90315
235.492.8997
30.171.944.917.10314
223.093.8527
9.571.923.291.75913
21598.3506
358.023.320.85312
20237.7216
56.511.511.56511
19Prime!          230.6636
53.205.650.23511
1875.1465
5.646.996.46510
1773.1885
5.356.556.53510
1630.9475
957.747.7599
1523.1115
534.141.4359
146.1544
37.877.8738
132.4314
5.912.1957
127533
567.7656
112433
59.2955
101843
34.0435
9Prime!          1793
32.2235
81753
Prime!          30.8035
7Prime!          712
5.1154
6602
3.6634
5Prime!          232
5553
4Prime!          192
Prime!          3833
3Prime!          51
332
2Prime!          21
91
111
Prime!          51
001
Prime!          31
up down
Case X = 4
One can find the regular numbers of the form n2+(n+4) at
%N n^2 + (n+4). under A027689.
The palindromic numbers of the form n2+(n+4) are categorised as follows :
%N n^2 + (n+4) is a palindrome. under A027716.
%N Palindromes of the form n^2 + (n+4). under A027717.
31200.000.000.00012
40.000.000.000.200.000.000.00423
3024.784.431.17111
614.268.028.494.820.862.41621
2921.572.321.96011
465.365.074.767.470.563.56421
2820.349.812.81411
414.114.881.585.188.411.41421
2720.102.030.40011
404.091.626.222.626.190.40421
2620.000.000.00011
400.000.000.020.000.000.00421
252.094.600.19410
4.387.349.974.799.437.83419
242.000.000.00010
4.000.000.002.000.000.00419
23205.705.7049
42.314.836.863.841.32417
22200.000.0009
40.000.000.200.000.00417
2125.882.3538
669.896.222.698.96615
2020.000.0009
400.000.020.000.00415
192.620.4867
6.866.949.496.68613
182.490.2667
6.201.427.241.02613
172.155.1397
4.644.626.264.46413
162.000.0007
4.000.002.000.00413
15218.3546
47.678.687.67411
14205.7046
42.314.341.32411
13202.8856
41.162.526.11411
12200.0006
40.000.200.00411
1120.6045
424.545.4249
1020.0005
400.020.0049
92.6134
6.830.3867
8Prime!          2.5514
6.510.1567
72.2344
4.992.9947
62.0004
4.002.0047
52613
68.3865
42193
48.1845
32003
40.2045
2202
4243
111
61
001
41
up down
Case X = 5
One can find the regular numbers of the form n2+(n+5) at
%N n^2 + (n+5). under A027690.
The palindromic numbers of the form n2+(n+5) are categorised as follows :
%N n^2 + (n+5) is a palindrome. under A027718.
%N Palindromes of the form n^2 + (n+5). under A027728.
282.237.334.99910
5.005.667.899.987.665.00519
27Prime!          106.617.6179
11.367.316.361.376.31117
2687.579.7538
7.670.213.223.120.76716
2585.125.9338
7.246.424.554.246.42716
247.569.2457
57.293.477.439.27514
234.063.8927
16.515.222.251.56114
222.775.3837
7.702.753.572.07713
212.724.7187
Prime!          7.424.090.904.24713
202.358.1507
5.560.873.780.65513
19107.4226
11.539.593.51111
18105.3226
11.092.829.01111
1728.1685
793.464.3979
1627.5485
758.919.8579
1522.4245
502.858.2059
1412.7525
162.626.2619
134.0674
16.544.5618
124.0124
16.100.1618
113.4624
11.988.9118
102.7514
7.570.7577
92303
53.1355
82243
50.4055
7Prime!          1273
16.2615
6742
5.5554
5262
7073
4122
1613
381
772
2Prime!          21
Prime!          112
111
Prime!          71
001
Prime!          51
up down
Case X = 6
One can find the regular numbers of the form n2+(n+6) at
%N n^2 + (n+6). under A027691.
The palindromic numbers of the form n2+(n+6) are categorised as follows :
%N n^2 + (n+6) is a palindrome. under A027729.
%N Palindromes of the form n^2 + (n+6). under A027721.
302.853.889.20310
8.144.683.585.853.864.41819
29288.888.8889
83.456.789.898.765.43817
28264.056.4349
69.725.800.600.852.79617
27148.278.1379
21.986.406.060.468.91217
2629.585.5088
875.302.313.203.57815
2529.190.7488
852.099.797.990.25815
2428.888.8888
834.567.878.765.43815
2316.353.5478
267.438.515.834.76215
2214.604.6578
213.296.020.692.31215
212.995.6317
8.973.808.083.79813
20Prime!          2.985.9437
8.915.858.585.19813
192.960.8167
8.766.434.346.67813
182.952.3167
8.716.172.716.17813
172.888.8887
8.345.676.765.43813
162.873.2337
8.255.470.745.52813
15Prime!          2.834.1017
8.032.131.312.30813
14292.2766
85.425.552.45811
13288.8886
83.456.565.43811
12264.3346
69.872.727.89611
11Prime!          146.7776
21.543.634.51211
1028.8885
834.545.4389
92.9464
8.681.8687
82.8884
8.343.4387
72.5504
6.505.0567
62.4854
6.177.7167
52883
83.2385
4282
8183
3252
6563
2242
6063
111
81
001
61
up down
Case X = 7
One can find the regular numbers of the form n2+(n+7) at
%N n^2 + (n+7). under A027692.
The palindromic numbers of the form n2+(n+7) are categorised as follows :
%N n^2 + (n+7) is a palindrome. under A027722.
%N Palindromes of the form n^2 + (n+7). under A027723.
263.000.000.00110
9.000.000.009.000.000.00919
252.817.818.39010
7.940.100.481.840.010.49719
242.673.533.18510
7.147.779.693.969.777.41719
23303.758.4589
92.269.201.110.296.22917
22300.000.0019
90.000.000.900.000.00917
2130.122.0988
907.340.818.043.70915
20Prime!          30.000.0018
900.000.090.000.00915
1927.515.1258
757.082.131.280.75715
183.000.0017
9.000.009.000.00913
171.762.1227
3.105.075.705.01313
16312.2086
97.474.147.47911
15300.0016
90.000.900.00911
14177.2376
31.413.131.41311
1330.0015
900.090.0099
12Prime!          27.2595
743.080.3479
11Prime!          3.0014
9.009.0097
102.7644
7.642.4677
9Prime!          1.7774
3.159.5137
8Prime!          3133
98.2895
73013
90.9095
62803
78.6875
52743
75.3575
41773
Prime!          31.5135
3Prime!          312
9993
2Prime!          172
Prime!          3133
111
91
001
Prime!          71
up down
Case X = 8
One can find the regular numbers of the form n2+(n+8) at
%N n^2 + (n+8). under A027693.
The palindromic numbers of the form n2+(n+8) are categorised as follows :
%N n^2 + (n+8) is a palindrome. under A027724.
%N Palindromes of the form n^2 + (n+8). under A027725.
15 ?
 ?
142.926.428.84910
8.563.985.811.185.893.65819
13287.010.9209
82.375.268.486.257.32817
122.997.3347
8.984.014.104.89813
112.990.3907
8.942.435.342.49813
102.837.8757
8.053.537.353.50813
92.834.6997
8.035.521.255.30813
8297.1196
88.279.997.28811
729.5705
874.414.4789
620.3775
415.242.5149
52.9354
8.617.1687
42.9254
8.558.5587
32853
81.5185
22023
41.0145
1Prime!          292
8783
001
81
up down
Case X = 9
One can find the regular numbers of the form n2+(n+9) at
%N n^2 + (n+9). under A027694.
The palindromic numbers of the form n2+(n+9) are categorised as follows :
%N n^2 + (n+9) is a palindrome. under A027726.
%N Palindromes of the form n^2 + (n+9). under A027727.
623.000.000.00010
9.000.000.003.000.000.00919
611.078.247.67310
1.162.618.045.408.162.61119
60989.943.0549
979.987.251.152.789.97918
59387.851.9939
150.429.168.861.924.05118
58349.101.6969
121.871.994.499.178.12118
57Prime!          346.313.5739
119.933.091.190.339.91118
56337.310.8989
113.778.642.246.877.31118
55302.382.6899
91.435.290.909.253.41917
54300.000.0009
90.000.000.300.000.00917
53128.536.6489
16.521.670.007.612.56117
52120.588.4369
14.541.571.017.514.54117
51109.666.0039
12.026.632.323.662.02117
5098.066.4398
9.617.026.556.207.16916
49Prime!          41.794.8538
1.746.809.779.086.47116
4835.409.4968
1.253.832.442.383.52116
4730.880.2748
953.591.353.195.35915
4630.592.6308
935.909.040.909.53915
4530.447.6958
927.062.161.260.72915
4430.000.0008
900.000.030.000.00915
4323.683.3928
560.903.080.309.06515
4213.220.9788
174.794.272.497.47115
4110.222.4568
104.498.616.894.40115
407.489.3927
56.091.000.019.06514
393.932.8067
15.466.966.966.45114
383.152.2907
9.936.935.396.39913
373.045.6797
9.276.163.616.72913
363.040.6047
9.245.275.725.42913
353.000.0007
9.000.003.000.00913
34Prime!          2.378.5077
5.657.297.927.56513
33989.1546
978.426.624.87912
32980.5606
961.498.894.16912
31963.3046
927.955.559.72912
30315.1656
99.329.292.39911
29300.0006
90.000.300.00911
28128.3486
16.473.337.46111
27125.2066
15.676.667.65111
26114.1416
13.028.282.03111
25Prime!          106.8016
Prime!          11.406.560.41111
2472.0625
5.193.003.91510
2330.6055
936.696.6399
2230.0005
900.030.0099
219.8544
97.111.1798
20Prime!          3.8814
15.066.0518
193.1444
9.887.8897
183.0004
9.003.0097
172.3674
5.605.0657
161.4134
1.997.9917
151.1414
Prime!          1.303.0317
149693
939.9396
134383
192.2916
12Prime!          3313
109.9016
113053
93.3395
103043
92.7295
93003
90.3095
81383
19.1915
7Prime!          312
1.0014
6302
9393
5222
5153
4Prime!          132
Prime!          1913
3Prime!          112
1413
291
992
111
Prime!          112
001
91




Further Topics Revealed


Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences

If you are interested in the primes of these numbers rather than the palindromes,
well look no further :

%N n^2 + n + 1 is prime. under A002384
%N Primes of form n^2 + n + 1. under A002383

%N n^2 + n + 3 is prime. under A027752
%N Primes of form n^2 + n + 3. under A027753

%N n^2 + n + 5 is prime. under A027754
%N Primes of form n^2 + n + 5. under A027755

%N n^2 + n + 7 is prime. under A027756
%N Primes of form n^2 + n + 7. or [n^2 + 3n+9] under A005471

%N n^2 + n + 9 is prime. under A027757
%N Primes of form n^2 + n + 9. under A027758

Two more formula's are included in the table. They are well known to prime-lovers.

n2 + n + 17 generates 16 primes for all n values 0 to 15 !

%N n^2 + n + 17 is prime. under A028823
%N n^2 + n + 17 is composite. under A007636
%N Primes of form n^2 + n + 17. under A007635

n2 + n + 41 generates 40 primes for all n values 0 to 39 !
%N n^2 + n + 41 is prime. under A002837
%N n^2 + n + 41 is composite. under A007634
%N Primes of form n^2 + n + 41. under A005846

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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( © All rights reserved ) - Last modified : September 30, 2015.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com