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 pattern
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

	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 %N n(n+3). under A028552. The palindromic numbers of the form n(n+3) are categorised as follows : %N n(n+3) is a palindrome. under A028553. %N Palindromes of the form n(n+3). under A028554.
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 %N n(n+4). [= m^2 – 4] under A028347. The palindromic numbers of the form n(n+4) are categorised as follows : %N n(n+4) is a palindrome. under A028555. %N Palindromes of the form n(n+4). under A028556.
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 %N n(n+5). under A028557. The palindromic numbers of the form n(n+5) are categorised as follows : %N n(n+5) is a palindrome. under A028558. %N Palindromes of the form n(n+5). under A028559.
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 %N n(n+6). under A028560. The palindromic numbers of the form n(n+6) are categorised as follows : %N n(n+6) is a palindrome. under A028561. %N Palindromes of the form n(n+6). under A028562.
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 %N n(n+7). under A028563. The palindromic numbers of the form n(n+7) are categorised as follows : %N n(n+7) is a palindrome. under A028564. %N Palindromes of the form n(n+7). under A028565.
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 %N n(n+8). under A028566. The palindromic numbers of the form n(n+8) are categorised as follows : %N n(n+8) is a palindrome. under A028567. %N Palindromes of the form n(n+8). under A028568.
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 %N n(n+9). under A028569. The palindromic numbers of the form n(n+9) are categorised as follows : %N n(n+9) is a palindrome. under A028570. %N Palindromes of the form n(n+9). under A028571.
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 : Neil Sloane's Integer Sequences That case X=2 with formula n(n+2) equalling n2–1 was already known to me. I was quite surprised to find an entry for case X = 4. %N n^2 – 4 denominators of wavelengths of hydrogen lines. under A028347. Numbers of the form n2–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 n2–Y. whereby Y = (X/2)2 or the square of the natural numbers. n(n+0) = n2–0 (trivial case) n(n+2) = n2–1 n(n+4) = n2–4 n(n+6) = n2–9 n(n+8) = n2–16 n(n+10) = n2–25 The variables at the left are the even numbers (0,2,4,6,8,10,...). The variables at the right are the squares of the natural numbers (0,1,4,9,16,25,...). Obviously all these numbers are composite. The intermediate Y variables can deliver primes. Here's a resumée : %N n^2 – 2. under A008865 %N n^2 – 2 is prime. under A028870 %N Primes of the form n^2 – 2. under A028871 %N n^2 – 3. under A028872 %N n^2 – 3 is prime. under A028873 %N Primes of the form n^2 – 3. under A028874 %N n^2 – 5. under A028875 %N n^2 – 5 is prime. under A028876 %N Primes of the form n^2 – 5. under A028877 %N n^2 – 6. under A028878 %N n^2 – 6 is prime. under A028879 %N Primes of the form n^2 – 6. under A028880 %N n^2 – 7. under A028881 %N n^2 – 7 is prime. under A028882 %N Primes of the form n^2 – 7. under A028883 %N n^2 – 8. under A028884 %N n^2 – 8 is prime. under A028885 %N Primes of the form n^2 – 8. under A028886 This last sequence is a subset of : %N Divisible only by primes congruent to 1 mod 8. under A004625 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.

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(All rights reserved) - Last modified : November 19, 2011.

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

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