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prime n.  and  zygodrome

First numbers that belong to both sets:

11, 11177, 11777, 22111, 22277, 22777, 33311, 33377, 44111, 44777, 55333, 55511, 77711, 77999, 88811, 1100777, 1110077, 1111333, 1115533, 1115599, 1116677, 1117799, 1133333, 1133777, 1144333, 1144477, 1144499, 1155577, 1166677, 1177711, 1177733, 1188899, 1199777, 1199999, 2200777, 2220077, 2221111, 2222333, 2225533, 2225599, 2227777, 2233999, 2244499, 2255111, 2255333, 2255999, 2266633, 2266777, 2277733, 2288833, 2299111, 2299933, 3311177, 3311999, 3331177, 3332233, 3333311, 3334411, 3335533, 3335599, 3336677, 3337777, 3338899, 3339977, 3344333, 3355577, 3355777, 3366611, 3366677, 3377111, 3377999, 3388877, 3388999, 3399911, 4400111, 4400777, 4411133, 4411333, 4422211, 4433333, 4433399, 4433999, 4440011, 4441111, 4441133, 4442233, 4443311, 4446677, 4448833, 4455533, 4499111, 4499933, 5511133, 5522233, 5522299, 5522999, 5550011, 5551111, 5552233, 5553311, 5554433, 5554499, 5555777, 5566111, 5566333, 5566777, 5588111, 5599777, 5599999, 6611777, 6622111, 6622211, 6622999, 6644111, 6655333, 6655577, 6660011, 6661111, 6662233, 6663311, 6667711, 6668833, 6668899, 6677777, 6688111, 6688999, 6699977, 7722277, 7733311, 7733377, 7766777, 7771177, 7774433, 7774499, 7775533, 7799111, 7799333, 7799999, 8800111, 8800777, 8800999, 8811199, 8822111, 8822299, 8833399, 8844499, 8855111, 8855533, 8866111, 8866633, 8877733, 8881111, 8881199, 8882233, 8884433, 8885533, 8886611, 8887777, 8889977, 8899333, 9911333, 9922111, 9922277, 9922777, 9933311, 9933377, 9955511, 9955577, 9977777, 9977999, 9991133, 9991199, 9994499, 9995533, 9995599, 9997711, 9997777, 11000111, 11111777, 11122333, 11133377, 11144477, 11155777, 11166611, 11177711, 11199977, 11333111, 11444999, 11666111, 11666999, 11777333, 11888111, 11888777, 11999111, 11999333, 11999777, 22200077, 22233311, 22244111, 22244333, 22244477, 22244777, 22255577, 22255999, 22288111, 22288999, 22299911, 22333111, 22333999, 22666333, 22888111, 22888777, 22999777, 33300011, 33322111, 33322277, 33322999, 33344777, 33344999, 33355111, 33355577, 33377711, 44000111, 44222333, 44333777, 44411177, 44411333, 44422111, 44422277, 44422333, 44444999, 44455577, 44455777, 44488111, 44488777, 44499977, 44555999, 44777333, 44888777, 44888999, 44999333, 44999777, 55000111, 55111333, 55444111, 55444999, 55500077, 55511999, 55522277, 55555111, 55555333, 55555777, 55566677, 55577111, 55588111, 55777333, 55888111, 55999777, 66622333, 66633377, 66644111, 66644411, 66644999, 66655999, 66677333, 66677777, 77000999, 77111777, 77111999, 77222777, 77333777, 77333999, 77444999, 77555999, 77711111, 77722111, 77722277, 77722999, 77733377, 77788811.

Some larger examples :