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de Polignac numbers
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 
204457974 2 
320333920687632185872 3 
40222875502229219 4 
539883798070010684419458911064105 5 
602068763020333902185872 6 
7425888880579950592426727920694427189426305 7 
801114748011151660111400701114053 8 
9678786892487283116763769094772878167824688568728780 9 
100980700094589103988370106844101064105 10 
11328534396425427481397145435969432918392570391784429349409381416418

A pictorial representation of the table above
motab
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.