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congruent n.  and  Sophie Germain prime

First numbers that belong to both sets:

5, 23, 29, 41, 53, 173, 191, 239, 293, 359, 431, 509, 653, 719, 743, 761, 911, 1013, 1031, 1103, 1223, 1229, 1439, 1511, 1559, 1583, 1733, 1901, 1973, 2039, 2063, 2069, 2141, 2273, 2351, 2399, 2543, 2549, 2693, 2741, 2903, 3023, 3359, 3389, 3413, 3593, 3623, 3761, 3821, 3863, 3911, 4271, 4349, 4373, 4391, 4481, 4733, 4793, 4871, 4919, 4943, 5039, 5231, 5279, 5303, 5333, 5399, 5501, 5639, 5711, 5741, 5849, 5903, 6053, 6101, 6173, 6263, 6269, 6551, 6581, 6983, 7079, 7103, 7151, 7349, 7541, 7823, 7901, 8069, 8093, 8111, 8663, 8693, 8741, 8951, 9029, 9221, 9293, 9479, 9629, 9791, 10061, 10253, 10271, 10589, 10613, 10709, 10733, 10781, 10799, 11369, 11471, 11519, 11549, 11783, 11813, 11831, 11909, 12101, 12119, 12263, 12653, 12671, 12791, 12821, 12959, 13229, 13463, 13901, 14159, 14303, 14321, 14621, 14669, 14741, 14783, 14831, 14879, 15101, 15173, 15269, 15569, 15629, 15773, 15791, 16253, 16301, 16421, 16493, 16673, 16823, 17159, 17183, 17333, 17351, 17669, 17981, 18149, 18191, 18341, 18461, 18773, 19301, 19319, 19373, 19391, 19559, 19661, 19709, 19751, 19919, 19991, 20063, 20693, 20759, 20789, 20879, 21149, 21221, 21341, 21383, 21701, 21893, 22013, 22079, 22133, 22271, 22343, 22349, 22469, 22541, 22751, 22853, 22943, 23279, 23669, 23909, 23981, 24239, 24509, 24551, 24749, 25229, 25601, 25703, 25799, 25919, 26111, 26189, 26501, 26573, 26879, 27143, 27479, 27551, 27581, 27743, 27773, 27893, 27983, 28559, 28751, 28901, 28949, 29021, 29453, 30269, 30389, 30671, 30773, 30983, 31151, 31253, 31319, 31469, 31799, 32141, 32159, 32381, 32573, 32633, 32789, 32933, 33023, 33053, 33119, 33191, 33461, 33479, 33521, 33623, 33749, 33773, 33941, 34253, 34319, 34439, 34631, 34949, 35069, 35111, 35573, 35831, 35933, 35999, 36191, 36383, 36479, 36629, 36791, 36821, 37013, 37181, 37253, 37853, 37871, 37991, 38039, 38183, 38189.

Some larger examples :