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twin primes
A member of a pair of consecutive primes whose difference is 2. more

The first 600 twin primes :
3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 179, 181, 191, 193, 197, 199, 227, 229, 239, 241, 269, 271, 281, 283, 311, 313, 347, 349, 419, 421, 431, 433, 461, 463, 521, 523, 569, 571, 599, 601, 617, 619, 641, 643, 659, 661, 809, 811, 821, 823, 827, 829, 857, 859, 881, 883, 1019, 1021, 1031, 1033, 1049, 1051, 1061, 1063, 1091, 1093, 1151, 1153, 1229, 1231, 1277, 1279, 1289, 1291, 1301, 1303, 1319, 1321, 1427, 1429, 1451, 1453, 1481, 1483, 1487, 1489, 1607, 1609, 1619, 1621, 1667, 1669, 1697, 1699, 1721, 1723, 1787, 1789, 1871, 1873, 1877, 1879, 1931, 1933, 1949, 1951, 1997, 1999, 2027, 2029, 2081, 2083, 2087, 2089, 2111, 2113, 2129, 2131, 2141, 2143, 2237, 2239, 2267, 2269, 2309, 2311, 2339, 2341, 2381, 2383, 2549, 2551, 2591, 2593, 2657, 2659, 2687, 2689, 2711, 2713, 2729, 2731, 2789, 2791, 2801, 2803, 2969, 2971, 2999, 3001, 3119, 3121, 3167, 3169, 3251, 3253, 3257, 3259, 3299, 3301, 3329, 3331, 3359, 3361, 3371, 3373, 3389, 3391, 3461, 3463, 3467, 3469, 3527, 3529, 3539, 3541, 3557, 3559, 3581, 3583, 3671, 3673, 3767, 3769, 3821, 3823, 3851, 3853, 3917, 3919, 3929, 3931, 4001, 4003, 4019, 4021, 4049, 4051, 4091, 4093, 4127, 4129, 4157, 4159, 4217, 4219, 4229, 4231, 4241, 4243, 4259, 4261, 4271, 4273, 4337, 4339, 4421, 4423, 4481, 4483, 4517, 4519, 4547, 4549, 4637, 4639, 4649, 4651, 4721, 4723, 4787, 4789, 4799, 4801, 4931, 4933, 4967, 4969, 5009, 5011, 5021, 5023, 5099, 5101, 5231, 5233, 5279, 5281, 5417, 5419, 5441, 5443, 5477, 5479, 5501, 5503, 5519, 5521, 5639, 5641, 5651, 5653, 5657, 5659, 5741, 5743, 5849, 5851, 5867, 5869, 5879, 5881, 6089, 6091, 6131, 6133, 6197, 6199, 6269, 6271, 6299, 6301, 6359, 6361, 6449, 6451, 6551, 6553, 6569, 6571, 6659, 6661, 6689, 6691, 6701, 6703, 6761, 6763, 6779, 6781, 6791, 6793, 6827, 6829, 6869, 6871, 6947, 6949, 6959, 6961, 7127, 7129, 7211, 7213, 7307, 7309, 7331, 7333, 7349, 7351, 7457, 7459, 7487, 7489, 7547, 7549, 7559, 7561, 7589, 7591, 7757, 7759, 7877, 7879, 7949, 7951, 8009, 8011, 8087, 8089, 8219, 8221, 8231, 8233, 8291, 8293, 8387, 8389, 8429, 8431, 8537, 8539, 8597, 8599, 8627, 8629, 8819, 8821, 8837, 8839, 8861, 8863, 8969, 8971, 8999, 9001, 9011, 9013, 9041, 9043, 9239, 9241, 9281, 9283, 9341, 9343, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9629, 9631, 9677, 9679, 9719, 9721, 9767, 9769, 9857, 9859, 9929, 9931, 10007, 10009, 10037, 10039, 10067, 10069, 10091, 10093, 10139, 10141, 10271, 10273, 10301, 10303, 10331, 10333, 10427, 10429, 10457, 10459, 10499, 10501, 10529, 10531, 10709, 10711, 10859, 10861, 10889, 10891, 10937, 10939, 11057, 11059, 11069, 11071, 11117, 11119, 11159, 11161, 11171, 11173, 11351, 11353, 11489, 11491, 11549, 11551, 11699, 11701, 11717, 11719, 11777, 11779, 11831, 11833, 11939, 11941, 11969, 11971, 12041, 12043, 12071, 12073, 12107, 12109, 12161, 12163, 12239, 12241, 12251, 12253, 12377, 12379, 12539, 12541, 12611, 12613, 12821, 12823, 12917, 12919, 13001, 13003, 13007, 13009, 13217, 13219, 13337, 13339, 13397, 13399, 13679, 13681, 13691, 13693, 13709, 13711, 13721, 13723, 13757, 13759, 13829, 13831, 13877, 13879, 13901, 13903, 13931, 13933, 13997, 13999, 14009, 14011, 14081, 14083, 14249, 14251, 14321, 14323, 14387, 14389, 14447, 14449, 14549, 14551, 14561, 14563, 14591, 14593, 14627, 14629, 14867, 14869, 15137, 15139, 15269, 15271, 15287, 15289, 15329, 15331, 15359, 15361, 15581, 15583, 15641, 15643, 15647, 15649, 15731, 15733, 15737, 15739, 15887, 15889, 15971, 15973, 16061, 16063, 16067, 16069, 16139, 16141, 16187, 16189, 16229, 16231, 16361, 16363, 16451, 16453, 16631, 16633, 16649, 16651, 16691, 16693, 16829, 16831, 16901, 16903, 16979, 16981, 17027, 17029, 17189, 17191, 17207, 17209, 17291.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 6849011 values, from 3 to 999999193).

n\r 0  1 
206849011 2 
3134245053424505 3 
40342450503424506 4 
512282376114212911412182283287 5 
6034245050103424505 6 
711370191684919137031613696066847081369270 7 
801712651017124920171185401712014 8 
9011412501140981111409811142274011422741141250 9 
100228237601141218010114212902283287 10 
111760830380905761122761038761497760144760786760872380267761549

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.