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 )

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 :

	Palindromic Quasi_Over_Squares of form n^2+(n+X)
	Case X = 0
See in-depth webpage about Palindromic Pronic Numbers
	Case X = 1
See in-depth webpage about Palindromic Quasi-Over-Squares
	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.
	> 1.238.795.000
46	528.342.929	9
	279.146.251.152.641.972	18
45	482.441.669	9
	232.749.964.469.947.232	18
44	223.314.956	9
	49.869.569.796.596.894	17
43	213.934.623	9
	45.768.023.132.086.754	17
42	200.000.001	9
	40.000.000.600.000.004	17
41	161.280.539	9
	26.011.412.421.411.062	17
40	92.610.852	8
	8.576.770.000.776.758	16
39	48.829.930	8
	2.384.362.112.634.832	16
38	29.287.052	8
	857.731.444.137.758	15
37	22.012.038	8
	484.529.838.925.484	15
36	21.855.108	8
	477.645.767.546.774	15
35	21.147.906	8
	447.233.949.332.744	15
34	20.000.001	8
	400.000.060.000.004	15
33	17.101.809	8
	292.471.888.174.292	15
32	14.842.995	8
	220.314.515.413.022	15
31	2.000.001	7
	4.000.006.000.004	13
30	1.651.754	7
	2.728.292.928.272	13
29	1.620.660	7
	2.626.540.456.262	13
28	501.474	6
	251.476.674.152	12
27	Prime!          219.083	6
	47.997.579.974	11
26	200.001	6
	40.000.600.004	11
25	89.777	5
	8.059.999.508	10
24	28.377	5
	805.282.508	9
23	20.993	5
	440.727.044	9
22	20.001	5
	400.060.004	9
21	16.780	5
	281.585.182	9
20	2.183	4
	4.767.674	7
19	2.106	4
	4.437.344	7
18	2.001	4
	4.006.004	7
17	1.654	4
	2.737.372	7
16	1.530	4
	2.342.432	7
15	1.424	4
	2.029.202	9
14	897	3
	805.508	6
13	Prime!          479	3
	229.922	6
12	292	3
	85.558	5
11	203	3
	41.414	5
10	201	3
	40.604	5
9	92	2
	8.558	4
8	50	2
	2.552	4
7	21	2
	464	3
6	15	2
	242	3
5	14	2
	212	3
4	6	1
	44	2
3	4	1
	22	2
2	Prime!          2	1
	8	1
1	1	1
	4	1
0	0	1
	2	1
	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.
	> 1.263.805.000
35	727.913.433	9
	529.857.966.669.758.925	18
34	726.714.741	9
	528.114.315.513.411.825	18
33	589.167.859	9
	347.118.766.667.811.743	18
32	549.199.524	9
	301.620.117.711.026.103	18
31	Prime!          309.230.287	9
	95.623.370.707.332.659	17
30	230.098.566	9
	52.945.350.305.354.925	17
29	177.865.385	9
	31.636.095.359.063.613	17
28	76.675.186	8
	5.879.084.224.809.785	16
27	62.127.180	8
	3.859.786.556.879.583	16
26	30.909.092	8
	955.371.999.173.559	15
25	18.886.025	8
	356.681.959.186.653	15
24	17.605.724	8
	309.961.535.169.903	15
23	5.492.899	7
	30.171.944.917.103	14
22	3.093.852	7
	9.571.923.291.759	13
21	598.350	6
	358.023.320.853	12
20	237.721	6
	56.511.511.565	11
19	Prime!          230.663	6
	53.205.650.235	11
18	75.146	5
	5.646.996.465	10
17	73.188	5
	5.356.556.535	10
16	30.947	5
	957.747.759	9
15	23.111	5
	534.141.435	9
14	6.154	4
	37.877.873	8
13	2.431	4
	5.912.195	7
12	753	3
	567.765	6
11	243	3
	59.295	5
10	184	3
	34.043	5
9	Prime!          179	3
	32.223	5
8	175	3
	Prime!          30.803	5
7	Prime!          71	2
	5.115	4
6	60	2
	3.663	4
5	Prime!          23	2
	555	3
4	Prime!          19	2
	Prime!          383	3
3	Prime!          5	1
	33	2
2	Prime!          2	1
	9	1
1	1	1
	Prime!          5	1
0	0	1
	Prime!          3	1
	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.
	length 21 fully scanned
26	24.784.431.171	11
	614.268.028.494.820.862.416	21
25	2.094.600.194	10
	4.387.349.974.799.437.834	19
24	2.000.000.000	10
	4.000.000.002.000.000.004	19
23	205.705.704	9
	42.314.836.863.841.324	17
22	200.000.000	9
	40.000.000.200.000.004	17
21	25.882.353	8
	669.896.222.698.966	15
20	20.000.000	9
	400.000.020.000.004	15
19	2.620.486	7
	6.866.949.496.686	13
18	2.490.266	7
	6.201.427.241.026	13
17	2.155.139	7
	4.644.626.264.464	13
16	2.000.000	7
	4.000.002.000.004	13
15	218.354	6
	47.678.687.674	11
14	205.704	6
	42.314.341.324	11
13	202.885	6
	41.162.526.114	11
12	200.000	6
	40.000.200.004	11
11	20.604	5
	424.545.424	9
10	20.000	5
	400.020.004	9
9	2.613	4
	6.830.386	7
8	Prime!          2.551	4
	6.510.156	7
7	2.234	4
	4.992.994	7
6	2.000	4
	4.002.004	7
5	261	3
	68.386	5
4	219	3
	48.184	5
3	200	3
	40.204	5
2	20	2
	424	3
1	1	1
	6	1
0	0	1
	4	1
	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.
	> 1.000.000.000
27	Prime!          106.617.617	9
	11.367.316.361.376.311	17
26	87.579.753	8
	7.670.213.223.120.767	16
25	85.125.933	8
	7.246.424.554.246.427	16
24	7.569.245	7
	57.293.477.439.275	14
23	4.063.892	7
	16.515.222.251.561	14
22	2.775.383	7
	7.702.753.572.077	13
21	2.724.718	7
	Prime!          7.424.090.904.247	13
20	2.358.150	7
	5.560.873.780.655	13
19	107.422	6
	11.539.593.511	11
18	105.322	6
	11.092.829.011	11
17	28.168	5
	793.464.397	9
16	27.548	5
	758.919.857	9
15	22.424	5
	502.858.205	9
14	12.752	5
	162.626.261	9
13	4.067	4
	16.544.561	8
12	4.012	4
	16.100.161	8
11	3.462	4
	11.988.911	8
10	2.751	4
	7.570.757	7
9	230	3
	53.135	5
8	224	3
	50.405	5
7	Prime!          127	3
	16.261	5
6	74	2
	5.555	4
5	26	2
	707	3
4	12	2
	161	3
3	8	1
	77	2
2	Prime!          2	1
	Prime!          11	2
1	1	1
	Prime!          7	1
0	0	1
	Prime!          5	1
	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.
	> 1.649.862.000
29	288.888.888	9
	83.456.789.898.765.438	17
28	264.056.434	9
	69.725.800.600.852.796	17
27	148.278.137	9
	21.986.406.060.468.912	17
26	29.585.508	8
	875.302.313.203.578	15
25	29.190.748	8
	852.099.797.990.258	15
24	28.888.888	8
	834.567.878.765.438	15
23	16.353.547	8
	267.438.515.834.762	15
22	14.604.657	8
	213.296.020.692.312	15
21	2.995.631	7
	8.973.808.083.798	13
20	Prime!          2.985.943	7
	8.915.858.585.198	13
19	2.960.816	7
	8.766.434.346.678	13
18	2.952.316	7
	8.716.172.716.178	13
17	2.888.888	7
	8.345.676.765.438	13
16	2.873.233	7
	8.255.470.745.528	13
15	Prime!          2.834.101	7
	8.032.131.312.308	13
14	292.276	6
	85.425.552.458	11
13	288.888	6
	83.456.565.438	11
12	264.334	6
	69.872.727.896	11
11	Prime!          146.777	6
	21.543.634.512	11
10	28.888	5
	834.545.438	9
9	2.946	4
	8.681.868	7
8	2.888	4
	8.343.438	7
7	2.550	4
	6.505.056	7
6	2.485	4
	6.177.716	7
5	288	3
	83.238	5
4	28	2
	818	3
3	25	2
	656	3
2	24	2
	606	3
1	1	1
	8	1
0	0	1
	6	1
	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.
	> 1.896.772.000
23	303.758.458	9
	92.269.201.110.296.229	17
22	300.000.001	9
	90.000.000.900.000.009	17
21	30.122.098	8
	907.340.818.043.709	15
20	Prime!          30.000.001	8
	900.000.090.000.009	15
19	27.515.125	8
	757.082.131.280.757	15
18	3.000.001	7
	9.000.009.000.009	13
17	1.762.122	7
	3.105.075.705.013	13
16	312.208	6
	97.474.147.479	11
15	300.001	6
	90.000.900.009	11
14	177.237	6
	31.413.131.413	11
13	30.001	5
	900.090.009	9
12	Prime!          27.259	5
	743.080.347	9
11	Prime!          3.001	4
	9.009.009	7
10	2.764	4
	7.642.467	7
9	Prime!          1.777	4
	3.159.513	7
8	Prime!          313	3
	98.289	5
7	301	3
	90.909	5
6	280	3
	78.687	5
5	274	3
	75.357	5
4	177	3
	Prime!          31.513	5
3	Prime!          31	2
	999	3
2	Prime!          17	2
	Prime!          313	3
1	1	1
	9	1
0	0	1
	Prime!          7	1
	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.
	length 20 fully scanned
13	2.926.428.849	10
	8.563.985.811.185.893.658	19
12	2.997.334	7
	8.984.014.104.898	13
11	2.990.390	7
	8.942.435.342.498	13
10	2.837.875	7
	8.053.537.353.508	13
9	2.834.699	7
	8.035.521.255.308	13
8	297.119	6
	88.279.997.288	11
7	29.570	5
	874.414.478	9
6	20.377	5
	415.242.514	9
5	2.935	4
	8.617.168	7
4	2.925	4
	8.558.558	7
3	285	3
	81.518	5
2	202	3
	41.014	5
1	Prime!          29	2
	878	3
0	0	1
	8	1
	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.
58	349.101.696	9
	121.871.994.499.178.121	18
57	Prime!          346.313.573	9
	119.933.091.190.339.911	18
56	337.310.898	9
	113.778.642.246.877.311	18
55	302.382.689	9
	91.435.290.909.253.419	17
54	300.000.000	9
	90.000.000.300.000.009	17
53	128.536.648	9
	16.521.670.007.612.561	17
52	120.588.436	9
	14.541.571.017.514.541	17
51	109.666.003	9
	12.026.632.323.662.021	17
50	98.066.439	8
	9.617.026.556.207.169	16
49	Prime!          41.794.853	8
	1.746.809.779.086.471	16
48	35.409.496	8
	1.253.832.442.383.521	16
47	30.880.274	8
	953.591.353.195.359	15
46	30.592.630	8
	935.909.040.909.539	15
45	30.447.695	8
	927.062.161.260.729	15
44	30.000.000	8
	900.000.030.000.009	15
43	23.683.392	8
	560.903.080.309.065	15
42	13.220.978	8
	174.794.272.497.471	15
41	10.222.456	8
	104.498.616.894.401	15
40	7.489.392	7
	56.091.000.019.065	14
39	3.932.806	7
	15.466.966.966.451	14
38	3.152.290	7
	9.936.935.396.399	13
37	3.045.679	7
	9.276.163.616.729	13
36	3.040.604	7
	9.245.275.725.429	13
35	3.000.000	7
	9.000.003.000.009	13
34	Prime!          2.378.507	7
	5.657.297.927.565	13
33	989.154	6
	978.426.624.879	12
32	980.560	6
	961.498.894.169	12
31	963.304	6
	927.955.559.729	12
30	315.165	6
	99.329.292.399	11
29	300.000	6
	90.000.300.009	11
28	128.348	6
	16.473.337.461	11
27	125.206	6
	15.676.667.651	11
26	114.141	6
	13.028.282.031	11
25	Prime!          106.801	6
	Prime!          11.406.560.411	11
24	72.062	5
	5.193.003.915	10
23	30.605	5
	936.696.639	9
22	30.000	5
	900.030.009	9
21	9.854	4
	97.111.179	8
20	Prime!          3.881	4
	15.066.051	8
19	3.144	4
	9.887.889	7
18	3.000	4
	9.003.009	7
17	2.367	4
	5.605.065	7
16	1.413	4
	1.997.991	7
15	1.141	4
	Prime!          1.303.031	7
14	969	3
	939.939	6
13	438	3
	192.291	6
12	Prime!          331	3
	109.901	6
11	305	3
	93.339	5
10	304	3
	92.729	5
9	300	3
	90.309	5
8	138	3
	19.191	5
7	Prime!          31	2
	1.001	4
6	30	2
	939	3
5	22	2
	515	3
4	Prime!          13	2
	Prime!          191	3
3	Prime!          11	2
	141	3
2	9	1
	99	2
1	1	1
	Prime!          11	2
0	0	1
	9	1

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.

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(All rights reserved) - Last modified : August 25, 2004.

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

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