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Palindromic Quasipronicsof the form n(n+x) | |||

n(n+1) n(n+2) |

Palindromic numbers are numbers which read the same from

**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 )

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 patternAnd then there is this astonishing infinite pattern where the multiplier n+7 is palindromic itself !

you get also a palindromic sequence !

19 + 26 – 1 = 44

219 + 226 – 1 = 444

2109 + 2116 – 1 = 4224

21009 + 21016 – 1 = 42024

210009 + 210016 – 1 = 420024

2100009 + 2100016 – 1 = 4200024

21000009 + 21000016 – 1 = 42000024

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 ?

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
The palindromic numbers of the form n(n+3) are categorised as follows : | ||

33 | 2.999.999.998 | 10 |

8.999.999.996.999.999.998 | 19 | |

32 | 2.946.920.844 | 10 |

8.684.342.469.642.434.868 | 19 | |

31 | 2.134.473.706 | 10 |

4.555.978.008.008.795.554 | 19 | |

30 | 299.999.998 | 9 |

89.999.999.699.999.998 | 17 | |

29 | 294.174.669 | 9 |

86.538.736.763.783.568 | 17 | |

28 | 294.127.328 | 9 |

86.510.885.958.801.568 | 17 | |

27 | 289.053.683 | 9 |

83.552.032.523.025.538 | 17 | |

26 | 212.325.206 | 9 |

45.081.993.739.918.054 | 17 | |

25 | 29.999.998 | 8 |

899.999.969.999.998 | 15 | |

24 | 28.862.883 | 8 |

833.066.101.660.338 | 15 | |

23 | 21.341.691 | 8 |

455.467.838.764.554 | 15 | |

22 | 9.443.423 | 7 |

89.178.266.287.198 | 14 | |

21 | 2.999.998 | 7 |

8.999.996.999.998 | 13 | |

20 | 2.861.419 | 7 |

8.187.727.277.818 | 13 | |

19 | 2.122.206 | 7 |

4.503.764.673.054 | 13 | |

18 | 636.776 | 6 |

405.485.584.504 | 12 | |

17 | 299.998 | 6 |

89.999.699.998 | 11 | |

16 | Prime! 212.671 | 6 |

45.229.592.254 | 11 | |

15 | 212.206 | 6 |

45.032.023.054 | 11 | |

14 | 63.701 | 5 |

4.058.008.504 | 10 | |

13 | 29.998 | 5 |

899.969.998 | 9 | |

12 | 29.369 | 5 |

862.626.268 | 9 | |

11 | 28.814 | 5 |

830.333.038 | 9 | |

10 | 2.998 | 4 |

8.996.998 | 7 | |

9 | 2.126 | 4 |

4.526.254 | 7 | |

8 | 671 | 3 |

452.254 | 6 | |

7 | 298 | 3 |

89.698 | 5 | |

6 | Prime! 211 | 3 |

45.154 | 5 | |

5 | 88 | 2 |

8.008 | 4 | |

4 | 66 | 2 |

4.554 | 4 | |

3 | 28 | 2 |

868 | 3 | |

2 | 8 | 1 |

88 | 2 | |

1 | 1 | 1 |

4 | 1 | |

0 | 0 | 1 |

0 | 1 | |

Case X = 4 | ||

One can find the regular numbers of the form n(n+4) at
The palindromic numbers of the form n(n+4) are categorised as follows : | ||

39 | 1.455.445.544 | 10 |

2.118.321.737.371.238.112 | 19 | |

38 | 1.106.902.513 | 10 |

1.225.233.177.713.325.221 | 19 | |

37 | 486.849.854 | 9 |

237.022.782.287.220.732 | 18 | |

36 | 282.442.347 | 9 |

79.773.680.508.637.797 | 17 | |

35 | 277.945.157 | 9 |

77.253.511.411.535.277 | 17 | |

34 | 223.674.495 | 9 |

50.030.280.608.203.005 | 17 | |

33 | Prime! 110.651.063 | 9 |

12.243.658.185.634.221 | 17 | |

32 | 85.559.327 | 8 |

7.320.398.778.930.237 | 16 | |

31 | 14.554.452 | 8 |

211.832.131.238.112 | 15 | |

30 | 8.308.388 | 7 |

69.029.344.392.096 | 14 | |

29 | 7.395.091 | 7 |

54.687.400.478.645 | 14 | |

28 | 5.040.882 | 7 |

25.410.511.501.452 | 14 | |

27 | 3.504.083 | 7 |

12.278.611.687.221 | 14 | |

26 | 2.705.669 | 7 |

7.320.655.560.237 | 13 | |

25 | 1.659.022 | 7 |

2.752.360.632.572 | 13 | |

24 | 281.349 | 6 |

79.158.385.197 | 11 | |

23 | 266.737 | 6 |

71.149.694.117 | 11 | |

22 | 262.808 | 6 |

69.069.096.096 | 11 | |

21 | 145.544 | 6 |

21.183.638.112 | 11 | |

20 | 73.491 | 5 |

5.401.221.045 | 10 | |

19 | 23.255 | 5 |

540.888.045 | 9 | |

18 | 11.223 | 5 |

126.000.621 | 9 | |

17 | 11.063 | 5 |

122.434.221 | 9 | |

16 | 8.459 | 4 |

71.588.517 | 8 | |

15 | 8.338 | 4 |

69.555.596 | 8 | |

14 | 5.412 | 4 |

29.311.392 | 8 | |

13 | 1.582 | 4 |

2.509.052 | 7 | |

12 | 1.452 | 4 |

2.114.112 | 7 | |

11 | Prime! 1.123 | 4 |

1.265.621 | 7 | |

10 | 524 | 3 |

276.672 | 6 | |

9 | Prime! 269 | 3 |

73.437 | 5 | |

8 | 235 | 3 |

56.165 | 5 | |

7 | 144 | 3 |

21.312 | 5 | |

6 | 44 | 2 |

2.112 | 4 | |

5 | 33 | 2 |

1.221 | 4 | |

4 | 21 | 2 |

525 | 3 | |

3 | 14 | 2 |

252 | 3 | |

2 | Prime! 7 | 1 |

77 | 2 | |

1 | 1 | 1 |

5 | 1 | |

0 | 0 | 1 |

0 | 1 | |

Case X = 5 | ||

One can find the regular numbers of the form n(n+5) at
The palindromic numbers of the form n(n+5) are categorised as follows : | ||

23 | 8.161.664.181 | 10 |

66.612.762.244.226.721.666 | 20 | |

22 | 249.281.986 | 9 |

62.141.509.790.514.126 | 17 | |

21 | 211.313.148 | 9 |

44.653.247.574.235.644 | 17 | |

20 | 79.585.891 | 8 |

6.333.914.444.193.336 | 16 | |

19 | 6.887.078 | 7 |

47.431.877.813.474 | 14 | |

18 | 2.233.097 | 7 |

4.986.733.376.894 | 13 | |

17 | 2.187.142 | 7 |

4.783.601.063.874 | 13 | |

16 | 2.040.402 | 7 |

4.163.250.523.614 | 13 | |

15 | 207.067 | 6 |

42.877.777.824 | 11 | |

14 | 82.566 | 5 |

6.817.557.186 | 10 | |

13 | 64.493 | 5 |

4.159.669.514 | 10 | |

12 | 25.116 | 5 |

630.939.039 | 9 | |

11 | 20.402 | 5 |

416.343.614 | 9 | |

10 | Prime! 2.207 | 4 |

4.881.884 | 7 | |

9 | 2.178 | 4 |

4.754.574 | 7 | |

8 | 2.067 | 4 |

4.282.824 | 7 | |

7 | 814 | 3 |

666.666 | 6 | |

6 | 203 | 3 |

42.224 | 5 | |

5 | 202 | 3 |

41.814 | 5 | |

4 | 24 | 2 |

696 | 3 | |

3 | 18 | 2 |

414 | 3 | |

2 | 6 | 1 |

66 | 2 | |

1 | 1 | 1 |

6 | 1 | |

0 | 0 | 1 |

0 | 1 | |

Case X = 6 | ||

One can find the regular numbers of the form n(n+6) at
The palindromic numbers of the form n(n+6) are categorised as follows : | ||

30 | 1.414.071.397 | 10 |

1.999.597.924.297.959.991 | 19 | |

29 | 782.357.262 | 9 |

612.082.890.098.280.216 | 18 | |

28 | 440.021.417 | 9 |

193.618.850.058.816.391 | 18 | |

27 | 272.515.513 | 9 |

74.264.706.460.746.247 | 17 | |

26 | 235.769.755 | 9 |

55.587.378.787.378.555 | 17 | |

25 | 235.680.755 | 9 |

55.545.419.691.454.555 | 17 | |

24 | 88.837.611 | 8 |

7.892.121.661.212.987 | 16 | |

23 | 84.890.503 | 8 |

7.206.398.008.936.027 | 16 | |

22 | 73.114.035 | 8 |

5.345.662.552.665.435 | 16 | |

21 | 23.975.475 | 8 |

574.823.545.328.475 | 15 | |

20 | 23.887.419 | 8 |

570.608.929.806.075 | 15 | |

19 | 14.529.486 | 8 |

211.106.050.601.112 | 15 | |

18 | 7.560.069 | 7 |

57.154.688.645.175 | 14 | |

17 | 7.153.679 | 7 |

51.175.166.157.115 | 14 | |

16 | 5.401.396 | 7 |

29.175.111.157.192 | 14 | |

15 | 2.436.099 | 7 |

5.934.592.954.395 | 13 | |

14 | 2.273.915 | 7 |

5.170.703.070.715 | 13 | |

13 | 1.731.746 | 7 |

2.998.954.598.992 | 13 | |

12 | 1.378.507 | 7 |

1.900.289.820.091 | 13 | |

11 | 877.173 | 6 |

769.437.734.967 | 12 | |

10 | 486.618 | 6 |

236.799.997.632 | 12 | |

9 | 8.541 | 4 |

72.999.927 | 8 | |

8 | Prime! 2.731 | 4 |

7.474.747 | 7 | |

7 | Prime! 863 | 3 |

749.947 | 6 | |

6 | 715 | 3 |

515.515 | 6 | |

5 | 273 | 3 |

76.167 | 5 | |

4 | Prime! 137 | 3 |

19.591 | 5 | |

3 | 22 | 2 |

616 | 3 | |

2 | Prime! 5 | 1 |

55 | 2 | |

1 | 1 | 1 |

7 | 1 | |

0 | 0 | 1 |

0 | 1 | |

Case X = 7 | ||

One can find the regular numbers of the form n(n+7) at
The palindromic numbers of the form n(n+7) are categorised as follows : | ||

39 | 900.000.002 | 9 |

810.000.009.900.000.018 | 18 | |

38 | 703.210.974 | 9 |

494.505.678.876.505.494 | 18 | |

37 | 665.012.044 | 9 |

442.241.023.320.142.244 | 18 | |

36 | 221.990.214 | 9 |

49.279.656.665.697.294 | 17 | |

35 | 210.000.009 | 9 |

44.100.005.250.000.144 | 17 | |

34 | 90.000.002 | 8 |

8.100.000.990.000.018 | 16 | |

33 | 29.857.886 | 8 |

891.493.565.394.198 | 15 | |

32 | 29.802.897 | 8 |

888.212.878.212.888 | 15 | |

31 | 28.867.722 | 8 |

833.345.575.543.338 | 15 | |

30 | 21.155.344 | 8 |

447.548.727.845.744 | 15 | |

29 | 21.152.824 | 8 |

447.442.111.244.744 | 15 | |

28 | 21.117.304 | 8 |

445.940.676.049.544 | 15 | |

27 | 21.037.924 | 8 |

442.594.393.495.244 | 15 | |

26 | 21.000.009 | 8 |

441.000.525.000.144 | 15 | |

25 | 9.000.002 | 7 |

81.000.099.000.018 | 14 | |

24 | 8.972.876 | 7 |

80.512.566.521.508 | 14 | |

23 | 2.219.779 | 7 |

4.927.434.347.294 | 13 | |

22 | 2.109.664 | 7 |

4.450.696.960.544 | 13 | |

21 | 2.100.009 | 7 |

4.410.052.500.144 | 13 | |

20 | 942.997 | 6 |

889.249.942.988 | 12 | |

19 | 900.002 | 6 |

810.009.900.018 | 12 | |

18 | 669.024 | 6 |

447.597.795.744 | 12 | |

17 | 292.967 | 6 |

85.831.713.858 | 11 | |

16 | 223.114 | 6 |

49.781.418.794 | 11 | |

15 | 210.009 | 6 |

44.105.250.144 | 11 | |

14 | 90.002 | 5 |

8.100.990.018 | 10 | |

13 | 29.696 | 5 |

882.060.288 | 9 | |

12 | 21.009 | 5 |

441.525.144 | 9 | |

11 | 9.002 | 4 |

81.099.018 | 8 | |

10 | 2.982 | 4 |

8.913.198 | 7 | |

9 | 2.972 | 4 |

8.853.588 | 7 | |

8 | 2.109 | 4 |

4.462.644 | 7 | |

7 | 902 | 3 |

819.918 | 6 | |

6 | 664 | 3 |

445.544 | 6 | |

5 | 219 | 3 |

49.494 | 5 | |

4 | 26 | 2 |

858 | 3 | |

3 | Prime! 19 | 2 |

494 | 3 | |

2 | 4 | 1 |

44 | 2 | |

1 | 1 | 1 |

8 | 1 | |

0 | 0 | 1 |

0 | 1 | |

Case X = 8 | ||

One can find the regular numbers of the form n(n+8) at
The palindromic numbers of the form n(n+8) are categorised as follows : | ||

49 | 699.999.996 | 9 |

489.999.999.999.999.984 | 18 | |

48 | 695.334.376 | 9 |

483.489.900.009.984.384 | 18 | |

47 | 287.876.058 | 9 |

82.872.627.072.627.828 | 17 | |

46 | 225.207.185 | 9 |

50.718.277.977.281.705 | 17 | |

45 | 182.901.103 | 9 |

33.452.814.941.825.433 | 17 | |

44 | 91.842.938 | 8 |

8.435.125.995.215.348 | 16 | |

43 | 91.050.784 | 8 |

8.290.245.995.420.928 | 16 | |

42 | 69.999.996 | 8 |

4.899.999.999.999.984 | 16 | |

41 | 69.692.736 | 8 |

4.857.078.008.707.584 | 16 | |

40 | 59.318.009 | 8 |

3.518.626.666.268.153 | 16 | |

39 | 57.524.753 | 8 |

3.309.097.667.909.033 | 16 | |

38 | 23.352.327 | 8 |

545.331.363.133.545 | 15 | |

37 | Prime! 19.782.199 | 8 |

391.335.555.533.193 | 15 | |

36 | Prime! 17.869.169 | 8 |

319.307.343.703.913 | 15 | |

35 | 17.646.619 | 8 |

311.403.303.304.113 | 15 | |

34 | 7.243.225 | 7 |

52.464.366.346.425 | 14 | |

33 | 6.999.996 | 7 |

48.999.999.999.984 | 14 | |

32 | 2.281.817 | 7 |

5.206.707.076.025 | 13 | |

31 | Prime Curios! 1.934.063 | 7 |

3.740.615.160.473 | 13 | |

30 | 896.478 | 6 |

803.679.976.308 | 12 | |

29 | 699.996 | 6 |

489.999.999.984 | 12 | |

28 | 597.883 | 6 |

357.468.864.753 | 12 | |

27 | 298.144 | 6 |

88.892.229.888 | 11 | |

26 | 283.778 | 6 |

88.892.229.888 | 11 | |

25 | 241.097 | 6 |

58.129.692.185 | 11 | |

24 | 228.175 | 6 |

52.065.656.025 | 11 | |

23 | 188.583 | 6 |

35.565.056.553 | 11 | |

22 | 93.844 | 5 |

8.807.447.088 | 10 | |

21 | 69.996 | 5 |

4.899.999.984 | 10 | |

20 | 57.489 | 5 |

3.305.445.033 | 10 | |

19 | 30.031 | 5 |

902.101.209 | 9 | |

18 | 28.314 | 5 |

801.909.108 | 9 | |

17 | 9.394 | 4 |

88.322.388 | 8 | |

16 | 6.996 | 4 |

48.999.984 | 8 | |

15 | 2.828 | 4 |

8.020.208 | 7 | |

14 | 2.235 | 4 |

5.013.105 | 7 | |

13 | 924 | 3 |

861.168 | 6 | |

12 | 707 | 3 |

505.505 | 6 | |

11 | 696 | 3 |

489.984 | 6 | |

10 | 294 | 3 |

88.788 | 5 | |

9 | 284 | 3 |

82.928 | 5 | |

8 | 225 | 3 |

52.425 | 5 | |

7 | 216 | 3 |

48.384 | 5 | |

6 | Prime! 173 | 3 |

31.313 | 5 | |

5 | 91 | 2 |

9.009 | 4 | |

4 | 88 | 2 |

8.448 | 4 | |

3 | 66 | 2 |

4.884 | 4 | |

2 | Prime! 3 | 1 |

33 | 2 | |

1 | 1 | 1 |

9 | 1 | |

0 | 0 | 1 |

0 | 1 | |

Case X = 9 | ||

One can find the regular numbers of the form n(n+9) at
The palindromic numbers of the form n(n+9) are categorised as follows : | ||

39 | 15.114.678.514 | 11 |

228.453.506.717.605.354.822 | 21 | |

38 | 1.665.597.364 | 10 |

2.774.214.593.954.124.772 | 19 | |

37 | 1.632.565.357 | 10 |

2.665.269.659.569.625.662 | 19 | |

36 | Prime! 1.534.664.347 | 10 |

2.355.194.671.764.915.532 | 19 | |

35 | 1.458.904.659 | 10 |

2.128.402.817.182.048.212 | 19 | |

34 | 1.417.320.729 | 10 |

2.008.798.061.608.978.002 | 19 | |

33 | 514.668.957 | 9 |

264.884.139.931.488.462 | 18 | |

32 | 497.665.874 | 9 |

247.671.326.623.176.742 | 18 | |

31 | 164.399.727 | 9 |

27.027.271.717.272.072 | 17 | |

30 | 151.698.472 | 9 |

23.012.427.772.421.032 | 17 | |

29 | 54.072.524 | 8 |

2.923.838.338.383.292 | 16 | |

28 | 52.030.352 | 8 |

2.707.157.997.517.072 | 16 | |

27 | 49.407.017 | 8 |

2.441.053.773.501.442 | 16 | |

26 | 48.675.319 | 8 |

2.369.287.117.829.632 | 16 | |

25 | 26.216.753 | 8 |

687.318.373.813.786 | 15 | |

24 | 26.211.438 | 8 |

687.039.717.930.786 | 15 | |

23 | 26.132.578 | 8 |

682.911.868.119.286 | 15 | |

22 | 25.234.083 | 8 |

636.759.171.957.636 | 15 | |

21 | 15.336.644 | 8 |

235.212.787.212.532 | 15 | |

20 | 15.019.614 | 8 |

225.588.939.885.522 | 15 | |

19 | 14.790.532 | 8 |

218.759.969.957.812 | 15 | |

18 | 14.674.184 | 8 |

215.331.808.133.512 | 15 | |

17 | 1.547.147 | 7 |

2.393.677.763.932 | 13 | |

16 | 487.619 | 6 |

237.776.677.732 | 12 | |

15 | 160.187 | 6 |

25.661.316.652 | 11 | |

14 | 16.997 | 5 |

289.050.982 | 9 | |

13 | 15.904 | 5 |

253.080.352 | 9 | |

12 | 5.392 | 4 |

29.122.192 | 8 | |

11 | 1.694 | 4 |

2.884.882 | 7 | |

10 | 1.664 | 4 |

2.783.872 | 7 | |

9 | 1.639 | 4 |

2.701.072 | 7 | |

8 | 258 | 3 |

68.886 | 5 | |

7 | 248 | 3 |

63.736 | 5 | |

6 | Prime! 167 | 3 |

29.392 | 5 | |

5 | Prime! 157 | 3 |

26.062 | 5 | |

4 | Prime! 137 | 3 |

20.002 | 5 | |

3 | 44 | 2 |

2.332 | 4 | |

2 | 12 | 2 |

252 | 3 | |

1 | Prime! 2 | 1 |

22 | 2 | |

0 | 0 | 1 |

0 | 1 |

Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
That case X=2 with formula n(n+2) equalling n Numbers of the form n ^{2}–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 n ^{2}–Y.whereby Y = (X/2) ^{2} or the square of the natural numbers.^{2}–0 (trivial case)
^{2}–1
^{2}–4
^{2}–9
^{2}–16
^{2}–25The variables at the left are the even numbers (0,2,4,6,8,10,...). Click here to view some of the author's [ |

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