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cyclic n.  and  modest n.

First numbers that belong to both sets:

13, 19, 23, 29, 59, 69, 79, 89, 103, 109, 133, 199, 209, 211, 233, 299, 311, 409, 411, 433, 499, 509, 511, 515, 533, 545, 599, 611, 699, 709, 721, 733, 763, 799, 809, 811, 899, 911, 933, 1003, 1009, 1033, 1037, 1099, 1111, 1133, 1199, 1333, 1339, 1399, 1433, 1463, 1499, 1527, 1633, 1699, 1733, 1799, 1933, 1957, 1999, 2003, 2027, 2033, 2045, 2055, 2099, 2111, 2177, 2333, 2369, 2399, 2427, 2533, 2545, 2555, 2599, 2633, 2699, 2733, 2833, 2863, 2899, 2933, 2987, 2999, 3009, 3037, 3055, 3133, 3193, 3199, 3233, 3299, 3333, 3499, 3545, 3563, 3599, 3799, 3899, 4009, 4027, 4037, 4045, 4099, 4111, 4199, 4277, 4299, 4333, 4399, 4499, 4699, 4799, 4899, 4963, 4999, 5009, 5015, 5027, 5045, 5099, 5111, 5199, 5333, 5399, 5599, 5663, 5677, 5699, 5899, 5999, 6009, 6037, 6081, 6199, 6299, 6333, 6499, 6599, 6799, 6899, 6999, 7009, 7021, 7027, 7037, 7099, 7199, 7259, 7333, 7499, 7599, 7699, 7777, 7799, 7899, 7999, 8009, 8027, 8099, 8111, 8299, 8333, 8399, 8599, 8699, 8899, 8999, 9037, 9199, 9299, 9499, 9599, 9799, 9899, 10009, 10011, 10015, 10027, 10033, 10037, 10099, 10111, 10135, 10303, 10333, 10909, 11027, 11033, 11099, 11111, 11333, 11407, 12037, 13027, 13037, 13039, 13111, 13117, 13333, 13481, 13999, 14037, 14111, 14189, 14333, 14777, 14999, 15027, 15037, 15045, 15185, 15999, 16027, 16111, 16333, 16999, 17027, 17111, 17153, 17333, 17999, 18037, 19027, 19037, 19057, 19111, 19171, 19333, 19703, 20003, 20009, 20011, 20027, 20033, 20099, 20101, 20135, 20303, 20333, 21037, 21259, 22027, 22037, 22111, 22297, 22333, 22999, 23027, 23069, 23207, 23333, 23851, 23999, 24027, 24333, 25027, 25037, 25045, 25111, 25555, 25999, 26111, 26333, 26351, 27037, 28111, 28189, 28259, 28333, 28777, 28999, 29111, 29261, 29333, 29999, 30011, 30045.

Some larger examples :