An odd number that cannot be written as the sum of a prime and a power of 2. more
The first 600 de Polignac numbers :
1,
127,
149,
251,
331,
337,
373,
509,
599,
701,
757,
809,
877,
905,
907,
959,
977,
997,
1019,
1087,
1199,
1207,
1211,
1243,
1259,
1271,
1477,
1529,
1541,
1549,
1589,
1597,
1619,
1649,
1657,
1719,
1759,
1777,
1783,
1807,
1829,
1859,
1867,
1927,
1969,
1973,
1985,
2171,
2203,
2213,
2231,
2263,
2279,
2293,
2377,
2429,
2465,
2503,
2579,
2669,
2683,
2789,
2843,
2879,
2909,
2983,
2993,
2999,
3029,
3119,
3149,
3163,
3181,
3187,
3215,
3239,
3299,
3341,
3343,
3353,
3431,
3433,
3505,
3539,
3637,
3643,
3665,
3697,
3739,
3779,
3817,
3845,
3877,
3967,
3985,
4001,
4013,
4063,
4151,
4153,
4195,
4229,
4271,
4311,
4327,
4503,
4543,
4567,
4573,
4589,
4633,
4649,
4663,
4691,
4717,
4781,
4811,
4813,
4841,
4843,
4855,
4889,
5077,
5099,
5125,
5143,
5303,
5323,
5405,
5467,
5557,
5609,
5617,
5729,
5731,
5737,
5755,
5761,
5771,
5917,
5923,
5951,
6001,
6021,
6065,
6073,
6119,
6161,
6173,
6193,
6247,
6269,
6283,
6403,
6433,
6449,
6463,
6509,
6521,
6535,
6539,
6547,
6637,
6659,
6673,
6731,
6757,
6791,
6821,
6853,
6869,
6883,
6941,
7109,
7151,
7169,
7177,
7199,
7267,
7289,
7297,
7319,
7331,
7343,
7379,
7387,
7389,
7393,
7405,
7417,
7431,
7517,
7535,
7547,
7583,
7603,
7747,
7753,
7783,
7799,
7807,
7811,
7813,
7841,
7867,
7901,
7913,
7961,
8023,
8031,
8087,
8107,
8111,
8141,
8159,
8257,
8287,
8363,
8387,
8399,
8411,
8429,
8467,
8527,
8563,
8587,
8621,
8669,
8719,
8789,
8831,
8849,
8861,
8873,
8887,
8915,
8921,
8923,
8929,
8981,
9101,
9115,
9239,
9307,
9371,
9391,
9431,
9457,
9473,
9517,
9521,
9557,
9569,
9581,
9613,
9641,
9787,
9809,
9907,
9929,
9941,
9959,
10001,
10007,
10021,
10027,
10061,
10079,
10121,
10199,
10235,
10237,
10253,
10327,
10357,
10379,
10391,
10409,
10447,
10451,
10483,
10511,
10513,
10553,
10607,
10619,
10697,
10753,
10777,
10781,
10873,
10949,
10963,
11015,
11023,
11039,
11069,
11081,
11083,
11105,
11137,
11141,
11171,
11207,
11219,
11227,
11231,
11239,
11279,
11285,
11317,
11335,
11347,
11411,
11435,
11437,
11533,
11541,
11549,
11579,
11593,
11627,
11695,
11729,
11743,
11771,
11789,
11801,
11857,
11909,
11921,
11993,
12007,
12131,
12191,
12203,
12223,
12233,
12239,
12251,
12371,
12373,
12401,
12427,
12431,
12479,
12517,
12595,
12671,
12727,
12731,
12733,
12749,
12791,
12805,
12877,
12881,
12929,
12941,
13001,
13083,
13091,
13093,
13099,
13147,
13169,
13217,
13285,
13297,
13351,
13393,
13409,
13451,
13469,
13589,
13603,
13619,
13679,
13735,
13799,
13841,
13859,
13897,
13901,
13961,
13973,
14009,
14021,
14023,
14039,
14047,
14051,
14077,
14081,
14101,
14107,
14141,
14143,
14227,
14231,
14249,
14279,
14303,
14347,
14375,
14381,
14383,
14407,
14437,
14459,
14467,
14473,
14489,
14531,
14533,
14585,
14605,
14611,
14639,
14681,
14765,
14809,
14879,
14917,
14921,
14975,
14981,
15013,
15037,
15041,
15043,
15059,
15071,
15101,
15113,
15119,
15121,
15127,
15149,
15161,
15187,
15217,
15223,
15247,
15299,
15349,
15359,
15373,
15401,
15419,
15521,
15551,
15607,
15641,
15701,
15719,
15779,
15787,
15809,
15853,
15869,
15943,
15957,
15997,
16013,
16025,
16027,
16031,
16109,
16117,
16165,
16177,
16181,
16213,
16361,
16405,
16409,
16499,
16507,
16543,
16559,
16601,
16629,
16645,
16727,
16739,
16753,
16769,
16783,
16787,
16849,
16865,
16867,
16973,
17021,
17039,
17047,
17077,
17083,
17089,
17113,
17137,
17147,
17229,
17257,
17269,
17305,
17327,
17339,
17369,
17371,
17411,
17429,
17437,
17467,
17489,
17519,
17557,
17579,
17593,
17651,
17669,
17735,
17759,
17767,
17773,
17827,
17849,
17861,
17887,
17909,
17921,
17957,
17977,
18033,
18089,
18103,
18155,
18209,
18211,
18307,
18359,
18391,
18427,
18487,
18517,
18551,
18565,
18607,
18611,
18613,
18637,
18685,
18719,
18787,
18817,
18845,
18869,
18881,
18889,
18895,
18897,
18899,
18911,
18959,
18971,
19007,
19057,
19093,
19117,
19135,
19139,
19163,
19177,
19259,
19273,
19319,
19345,
19357,
19361,
19379,
19483,
19583,
19631,
19649,
19709,
19807,
19819,
19889,
19949,
19961,
20113,
20141,
20143,
20195,
20201,
20287,
20309,
20321,
20323.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 4457974 values, from 1 to 99999937).
n\r | 0 | 1 |
2 | 0 | 4457974 | 2 |
3 | 203339 | 2068763 | 2185872 | 3 |
4 | 0 | 2228755 | 0 | 2229219 | 4 |
5 | 398837 | 980700 | 1068441 | 945891 | 1064105 | 5 |
6 | 0 | 2068763 | 0 | 203339 | 0 | 2185872 | 6 |
7 | 425888 | 880579 | 950592 | 426727 | 920694 | 427189 | 426305 | 7 |
8 | 0 | 1114748 | 0 | 1115166 | 0 | 1114007 | 0 | 1114053 | 8 |
9 | 67878 | 689248 | 728311 | 67637 | 690947 | 728781 | 67824 | 688568 | 728780 | 9 |
10 | 0 | 980700 | 0 | 945891 | 0 | 398837 | 0 | 1068441 | 0 | 1064105 | 10 |
11 | 328534 | 396425 | 427481 | 397145 | 435969 | 432918 | 392570 | 391784 | 429349 | 409381 | 416418 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.